Construction Methods for Edge-Antimagic Labelings of Graphsprr.hec.gov.pk/jspui/bitstream/123456789/296/1/270S.pdf · 2018-07-17 · Graph labelings provide useful mathematical models

Construction Methods

for Edge-Antimagic Labelings of Graphs

Name: Muhammad Kashif Shafiq

Year of Admission: 2005

Registration No.: 24–GCU–PHD–SMS–05

Abdus Salam School of Mathematical Sciences

GC University, Lahore, Pakistan

Construction Methods

for Edge-Antimagic Labelings of Graphs

Submitted to


GC University Lahore, Pakistan

in the partial fulfillment of the requirements for the award of degree of

Doctor of Philosophy

in

Mathematics

By

Name: Muhammad Kashif Shafiq

Year of Admission: 2005

Registration No.: 24–GCU–PHD–SMS–05


GC University, Lahore, Pakistan

i

DECLARATION

I, Mr. Muhammad Kashif Shafiq Registration No. 24–GCU–PHD–SMS–05

student at Abdus Salam School of Mathematical Sciences GC University in

the subject of Mathematics, hereby declare that the matter printed in this thesis

titled

“Construction Methods for Edge-Antimagic Labelings of Graphs”

is my own work and that

(i) I am not registered for the similar degree elsewhere contemporaneously.

(ii) No direct major work had already been done by me or anybody else on this

topic; I worked on, for the Ph. D. degree.

(iii) The work, I am submitting for the Ph. D. degree has not already been submit-

ted elsewhere and shall not in future be submitted by me for obtaining similar

degree from any other institution.

Dated: ———————————— ————————————

Signature of Deponent

ii

RESEARCH COMPLETION CERTIFICATE

Certified that the research work contained in this thesis titled

“Construction Methods for Edge-Antimagic Labelings of Graphs”

has been carried out and completed by Mr. Muhammad Kashif Shafiq

Registration No. 24–GCU–PHD–SMS–05 under my supervision.

———————————— ————————————

Date Supervisor

Dr. Martin Baca

Submitted Through

Prof. Dr. A. D. Raza Choudary ———————————–

Director General Controller of Examination

Abdus Salam School of Mathematical Sciences GC University, Lahore

GC University, Lahore, Pakistan Pakistan

iii

Acknowledgement

In the name of Allah, the most Gracious and the most Merciful. All praise and

glory to Allah Almighty for His blessings upon me to finish my thesis.

I would like to thank Prof. Dr. Alla Ditta Raza Choudary for giving me

the opportunity to work in a professional research environment and under the super-

vision of world known professors. I also thank him for his help and encouragement

throughout the period of my studies.

I express my sincere gratitude to my research supervisor, Professor Dr. Mar-

tin Baca, for his quality supervision and constant support during this research. He

has been a role model as a teacher and as a researcher for me. This thesis would

not have been possible without his guidance. I am highly indebted to him for his af-

fectionate attitude, granting me great flexibility. Special thanks to my co-supervisor

Dr. Andrea Fenovcıkova for her guidance during my research and giving me valuable

suggestions in completing this thesis.

I am also thankful to my friends and colleagues of Batch III for giving me the

motivation to improve my academic competence. I am specially thankful to my

graph theory fellows (Mr. Abdul Qudair Baig, Mr. Ali Ahmad, Mr. Gohar Ali, Mr.

Muhammad Imran, Mr. Syed Ahtsham ul Haq Bokhary and Mrs. Fozia Bashir),

Dr. Asim Naseem, Dr. Hani Shakir, and Miss. Saima Parveen, for their meticulous

discussions on the subject with me.

Of course, I am grateful to my parents for their kindness, love, prayers, and

the great amount of confidence they have in me. I will never forget my sisters,

brothers, and my nephew Abdullah who always proved themselves to be a source of

encouragement for me during my study. Their fascinating smiles and warm welcome

iv

always remove my tiredness when I enter the house at evening.

I am also very thankful to my cousin Mr. Umair Khan, for reading this thesis

and making some corrections.

Finally, I wish to thank the following: All the foreign professors at ASSMS as

they made ASSMS a real place of learning and innovation; Mr. Syed Numan Jaffery,

Mr. Shaukat Ali, Mr. Zahid Nasir and Mr. Awais Naeem for providing a comfortable

work environment.

v

To my parents, Danish and Abdullah,

for their love, support and patience.

vi

Abstract

A labeling of a graph is a mapping that carries some set of graph elements

into numbers (usually positive integers). An (a, d)-edge-antimagic total labeling of

a graph, with p vertices and q edges, is a one-to-one mapping that takes the vertices

and edges into the integers 1, 2, . . . , p + q, so that the sums of the label on the edges

and the labels of their end vertices form an arithmetic progression starting at a and

having difference d. Such a labeling is called super if the p smallest possible labels

appear at the vertices.

This thesis deals with the existence of super (a, d)-edge-antimagic total labelings

of regular graphs and disconnected graphs.

We prove that every even regular graph and every odd regular graph, with a 1-

factor, admits a super (a, 1)-edge-antimagic total labeling. We study the super (a, 2)-

edge-antimagic total labelings of disconnected graphs and present some necessary

conditions for the existence of (a, d)-edge-antimagic total labelings for d even.

The thesis is also devoted to the study of edge-antimagicness of trees. We use

the connection between graceful labelings and edge-antimagic labelings for generating

large classes of edge-antimagic total trees from smaller graceful trees.

vii

List of publications arising from this thesis

[1] M. Baca, P. Kovar, A. Semanicova-Fenovcıkova and M.K. Shafiq, On Super

(a, 1)-edge-antimagic total labeling of regular graphs, Discrete Mathematics,

in press, doi:10.1016/j.disc.2009.04.011.

[2] M. Baca, A. Semanicova-Fenovcıkova and M.K. Shafiq, A method to generate

large classes of edge-antimagic trees, Utilitas Mathematica, in press.

[3] M. Baca, F.A. Muntaner-Batle, A. Semanicova-Fenovcıkova and M.K. Shafiq,

On super (a, 2)-edge-antimagic total labeling of disconnected graphs, Ars Com-

binatoria, in press.

Further publication produced during my PhD candidature

[1] M.K. Shafiq, G. Ali and R. Simanjuntak, Distance magic labelings of a union

of graphs, AKCE J. Graphs. Combin., 6, No. 1 (2009), 191–200.

viii

Contents

Introduction 1

1 Basic terminology and definitions 5

1.1 Graph theoretical terminology . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Antimagic labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Vertex-antimagic total labelings . . . . . . . . . . . . . . . . . 11

1.2.2 Edge-antimagic total labelings . . . . . . . . . . . . . . . . . . 13

1.3 Graceful labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.1 α-labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Super (a, d)-edge-antimagic total labelings for d even 21

2.1 Super (a, 2)-edge-antimagic total labelings for the disjoint union of

graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Conditions for non-existence of (super) (a, d)-edge-antimagic total la-

belings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Super (a, d)-edge-antimagic total labelings for d odd 33

3.1 Super (a, 1)-edge-antimagic total labelings for regular graphs . . . . . 34

3.2 Super (a, 1)-edge-antimagic total labelings for non-regular graphs . . 39

ix

4 Generating large classes of edge-antimagic total trees 43

4.1 Connections between α-labelings and edge-

antimagic vertex labelings for trees . . . . . . . . . . . . . . . . . . . 44

4.2 Construction of α-tree from smaller graceful trees . . . . . . . . . . . 46

4.3 Generating edge-antimagic trees . . . . . . . . . . . . . . . . . . . . . 50

4.4 Certain families of super (a, d)-edge-antimagic total trees . . . . . . . 52

Appendix – Graph-theoretic symbols 57

Bibliography 58

x

List of Figures

1.1 Wheel W7 and friendship graph F4. . . . . . . . . . . . . . . . . . . . 7

1.2 Fan F5 and pachachute graph P5,7. . . . . . . . . . . . . . . . . . . . 8

1.3 Graphs P (5, 2) and P (7, 3). . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Graph C3 × P6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Antimagic labeling of a tree. . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 (15, 2)-antimagic labeling of prism C5 × P2. . . . . . . . . . . . . . . . 11

1.7 Super (20, 2)-VAT labeling of cycle C7. . . . . . . . . . . . . . . . . . 12

1.8 (5, 1)-EAV labeling of cycle C7. . . . . . . . . . . . . . . . . . . . . . 14

1.9 (19, 2)-EAT labeling of cycle C8. . . . . . . . . . . . . . . . . . . . . . 15

1.10 Graceful labeling of a tree. . . . . . . . . . . . . . . . . . . . . . . . . 18

1.11 α-labeling of a caterpillar. . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Super (12, 2)-EAT labeling of 2P3. . . . . . . . . . . . . . . . . . . . . 22

2.2 Edge-magic total labeling of 2P3. . . . . . . . . . . . . . . . . . . . . 22

4.1 (a, 1)-EAV labeling of a tree that is not an α-tree. . . . . . . . . . . . 44

4.2 Cycle C6 and Star K1,5. . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Tree C6 v©K1,5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4 Graceful labelings of lobsters. . . . . . . . . . . . . . . . . . . . . . . 52

xi

4.5 Graceful labeling of a symmetrical tree ST . . . . . . . . . . . . . . . 53

4.6 α-labeling of P2 v©ST . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7 α-labeling of P4 v©ST . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.8 (3, 2)-EAV labeling of P4 v©ST . . . . . . . . . . . . . . . . . . . . . . 55

xii

1

Introduction

A graph G is an ordered pair of sets G = (V, E), where V is a set of entities,

typically called points, or nodes, or vertices. The set of points V can be used to

represent all kinds of objects and the second set, E, represents a binary relation

between the points. The elements of the set E are usually called edges.

A graph can be labeled or unlabeled. In this thesis, we are interested in labeled

graphs. In many labeled graphs, the labels are used for identification only. The kind

of labeling we are interested in can serve dual purposes, it means a labeling can be

used to identify vertices and edges and also to signify some additional properties,

depending on the particular labeling.

Graph labelings provide useful mathematical models for a wide range of appli-

cations, such as data security, cryptography (secret sharing schemes), astronomy,

various coding theory problems, communication networks, mobile telecommunica-

tion systems, bioinformatics and x-ray crystallography. More detailed discussions

about applications of graph labelings can be found in Bloom and Golomb’s papers

[30] and [31].

Many studies in graph labeling refer to Rosa’s research in 1967 [82]. Rosa in-

troduced a function f from the set of vertices of a graph G to the set of integers

0, 1, 2, . . . , q, where q is the number of edges in G, so that each edge xy is assigned

the label |f(x)−f(y)|, with all labels distinct. Rosa called this labeling β-valuation.

Independently, Golomb [48] studied the same type of labeling and called this labeling

graceful labeling. The graceful labeling was broadly popularized in a paper by Gard-

ner in 1972 [47], mainly for its connection to the Ringel’s conjecture, which asserts

that every tree of size q decomposes the complete graph K2q+1. Ringel’s conjecture

2

can be derived by the Kotzig’s graceful conjecture, which asserts that every tree is

graceful.

Although Erdos proved in an unpublished paper that almost all graphs are not

graceful, many particular families of graphs have been proved to admit graceful

labelings. Among the trees known to be graceful are caterpillars [82], trees with at

most 4 end-vertices [58], trees with diameter at most 5 [57] and trees with at most

27 vertices [3].

In 1963 Sedlacek [84] published a paper about another kind of graph labeling. He

called the labeling magic. His definition was motivated by the magic square notion

in number theory. A magic labeling is a function from the set of edges of a graph

G into the non-negative real numbers, so that the sums of the edge labels around

any vertex in G are all the same. Stewart [90] called magic labeling supermagic if

the set of edge labels consisted of consecutive integers. Motivated by Sedlacek’s and

Stewart’s research, many new related definitions have been proposed and new results

have been found.

Graph labeling is an injective mapping from elements of a graph (can be vertices,

edges or a combination) to a set of numbers (usually positive integers). If the domain

of the mapping is the set of vertices or the set of edges then the labeling is called

vertex labeling or edge labeling, respectively. If the domain of the mapping is the set

of vertices and edges then the labeling is called total labeling. The mapping usually

produces partial sums of the labeled elements of the graph. The partial sums will

be either a set of vertex weights, obtained for each vertex by adding all the labels of

a vertex and its adjacent edges, or a set of edge weights, obtained for each edge by

adding the labels of an edge and its endpoints.

One of the situations that we are particularly interested in is when all the edge

3

weights or all the vertex weights are the same. In such a case we call the labeled

graph edge-magic or vertex-magic, respectively. Edge-magic and vertex-magic graphs

are described in the pioneering book by Wal Wallis [96].

Another situation that is of interest is when all the edge weights or all the vertex

weights are different. In such a case we call the labeled graph edge-antimagic or

vertex-antimagic, respectively. The study of these graphs was motivated by Hartsfield

and Ringel [56], who considered labeling uniquely the edges of a graph containing

q edges using the integers 1, 2, . . . , q, and evaluating partial sums of labels at the

vertices of the graph. If all the vertex weights are different then they call the graph

antimagic. Hartsfield and Ringel propose the conjecture that every connected graph

different from K2 is antimagic. This conjecture is still open. Even if we restrict

ourselves to trees, it is not known whether the conjecture is true.

While many researchers studied the properties of magic and antimagic labelings,

other researchers examined their applications. Kalantari, Khosrovshahi and Mitchell

in [60] and [73] tried to find applications of magic labeling in optimization theory,

especially for the traveling salesmen problem. Baskoro, Simanjuntak and Adithia

[26], [27] proposed a secret sharing scheme construction using edge-magic labeling.

Based on Bloom and Golomb’s results [30], [31], Wallis [96] proposed edge-magic

total labeling for assigning addresses of communication networks and radar pulse

codes. Recently, Hartnell and Rall [55] proposed a game based on vertex-magic edge

labeling.

4

Outline of the thesis

In Chapter 1 we will introduce some basic definitions and notations about graph

theory that are used through out the thesis and we will give an overview of vertex-

antimagic, edge-antimagic and graceful labelings and certain known results in these

research areas.

In Chapter 2 we will mainly investigate the existence of super (a, 2)-EAT labelings

for disconnected graphs. We will concentrate on the following problem: If a graph

G is (super) (a, 2)-EAT, is the disjoint union of m copies of the graph G, denoted

by mG, (super) (a, 2)-EAT as well? We will also present some necessary conditions

for the existence of (a, d)-edge-antimagic total labelings for d even.

In Chapter 3 we will deal with the existence of super (a, 1)-EAT labelings of

regular graphs. We will also give some constructions of non-regular super (a, 1)-EAT

graphs. The constructions do not require the graph to be connected.

In Chapter 4 we will use the connection between graceful labelings and edge-anti-

magic labelings for generating large classes of edge-antimagic total trees from smaller

graceful trees.

At the end of the thesis, in Appendix A, the list of graph-theoretic symbols used

in the thesis is given.

Chapter 1

Basic terminology and definitions

1.1 Graph theoretical terminology

In this section we will give the basic definitions, notation and terminology in the

graph theory. For other concepts which are not explicitly given in this section see

[14], [38], [96] and [99].

A graph G = G(V, E) consists of two sets V (G) and E(G), where V = V (G) is

a set of vertices and E = E(G) is a set of edges. We say that a graph G has order p

and size q if |V (G)| = p and |E(G)| = q. A graph G that has order p and size q is

sometimes called a (p, q)-graph. A graph is said to be finite if the order p is finite.

A graph is called simple if there are neither loops (an edge that has both endpoints

the same) nor multiple edges (more than one edge between two vertices). From now

on, every graph mentioned in the thesis is a simple finite graph.

There are many ways to represent a graph. However, traditionally a graph is

represented by a diagram. A dot represents a vertex and a curve, usually a line

segment, represents an edge.

5

Chapter 1. Basic terminology and definitions 6

In a graph G, a vertex x is said to be adjacent to the vertex y if there is an edge

e between x and y, that is e = xy. The vertices x and y are called the endpoints

of an edge xy. The vertex y is then called a neighbor of x, or we say that x and y

are incident with the edge e. The set of all neighbors of the vertex x in a graph G

is denoted by the symbol N(x). The number of neighbors of x is called the degree

of a vertex x, denoted by d(x). If d(x) = 0, it means that x is not adjacent to any

other vertex, then x is called an isolated vertex. A vertex of degree 1 is called an end

vertex or a pendant vertex or a leaf.

The minimum degree of a graph G is denoted by δ = δ(G) and the maximum

degree of a graph G is denoted by ∆ = ∆(G). If every vertex in a graph G has the

same degree r, that is δ = ∆ = r, then G is called a regular graph of degree r, or

an r-regular graph.

A graph H is a subgraph of G, denoted by H ⊆ G, if every vertex of H is a vertex

of G and every edge of H is an edge of G. In other words, V (H) ⊆ V (G) and

E(H) ⊆ E(G). We say that a subgraph H is a spanning subgraph, or a factor, of

G if H contains all vertices of G, i.e. V (H) = V (G). A k-factor is a factor that is

k-regular, that is, every vertex in the factor has a degree k.

By Pn we denote the path on n vertices. The cycle on n vertices is denoted

by the symbol Cn. A graph is bipartite if it is possible to categorize its vertices

into two partite sets, such that there are no edges between vertices in the same set.

A bipartite graph is complete if each vertex in one set is adjacent to all vertices in

the second set. The symbol Km,n denotes the complete bipartite graph with partite

sets of cardinalities m and n. A complete bipartite graph K1,n is a star on n + 1

vertices.

A complete graph Kn of order n is a graph in which every two distinct vertices


are adjacent. Kn is an (n− 1)-regular graph.

A wheel Wn, n ≥ 3, is a graph obtained by joining all vertices of cycle Cn to

a further vertex called the centre. Thus, Wn contains n + 1 vertices and 2n edges.

A friendship graph Fn, n ≥ 1, is a set of n triangles having a common central vertex,

and otherwise disjoint. The friendship graph Fn contains 2n + 1 vertices and 3n

edges. A fan Fn, n ≥ 2, is a graph obtained by joining all vertices of a path Pn

to a further vertex, called the centre. Thus, Fn contains n + 1 vertices and 2n − 1

edges. The parachute graph Pm,n is obtained from the wheel Wm+n by deleting n

consecutive spokes. The wheel W7 and the friendship graph F4 are displayed in

Figure 1.1. Figure 1.2 depicts the fan F5 and the pachachute graph P5,7.

Figure 1.1: Wheel W7 and friendship graph F4.

A generalized Petersen graph P (n,m), n ≥ 3 and 1 ≤ m ≤ b(n− 1)/2c, consists

of an outer n-cycle y0y1 . . . yn−1, a set of n spokes yixi, 0 ≤ i ≤ n − 1, and n edges

xixi+m, 0 ≤ i ≤ n − 1, with indices taken modulo n. The standard Petersen graph

is the instance P (5, 2). By definition P (n, m) is a 3-regular graph with 2n vertices

and 3n edges. The generalized Petersen graph was introduced by Watkins in [98].

The standard Petersen graph P (5, 2) and the generalized Petersen graphs P (7, 3) are


Figure 1.2: Fan F5 and pachachute graph P5,7.

depicted in Figure 1.3.

Figure 1.3: Graphs P (5, 2) and P (7, 3).

A Cartesian product of two graphs G and H, denoted by G × H, is the graph

with the vertex set V (G)× V (H), where two vertices (x, x′) and (y, y′) are adjacent

if and only if x = y and x′y′ ∈ E(H) or x′ = y′ and xy ∈ E(G). Figure 1.4 shows the

Cartesian product of the cycle C3 and the path P6. A prism Dn, n ≥ 3, is a 3-regular

graph which can be defined as the Cartesian product Cn × P2.

A labeling of a graph G is any mapping that sends some set of graph elements to

a set of numbers, usually the non-negative integers. If the domain is the vertex-set


Figure 1.4: Graph C3 × P6.

or the edge-set, the labeling is called vertex labeling or edge labeling, respectively.

Moreover, if the domain is V (G) ∪ E(G) then the labeling is called total labeling.

There are also many ways, not only which graph elements we label but also which

conditions must be satisfied. This is related to the weight of the graph element. Under

the vertex-weight we usually understand the sum of labels of all incident edges and

of the vertex label, if it is present. Analogously, the edge-weight is usually equal to

the sum of the edge label, if the edges are labeled, and the labels of its endpoints.

If the weights are the same we obtain so called magic-type labelings. On the other

hand, if the weights are different we have some variations of antimagic-type labeling.

1.2 Antimagic labelings

Hartsfield and Ringel in [56] introduced the concept of an antimagic labeling. In

their terminology, a (p, q) graph G is called antimagic if its edges are labeled with

labels 1, 2, . . . , q in such a way that all vertex-weights are pairwise distinct, where

a vertex-weight of vertex x is the sum of labels of all edges incident with x. Figure

1.5 gives an example of antimagic labeling of a small tree on 8 vertices.


5

7

1

2

36

4

Figure 1.5: Antimagic labeling of a tree.

Among the graphs known to be antimagic are paths, cycles, complete graphs and

wheels. It is easy to see that K2 is not antimagic. In fact, Hartsfield and Ringel

[56] conjecture that every connected graph, except K2, is antimagic. Alon, Kaplan,

Lev, Roditty and Yuster [5] used several probabilistic tools and some techniques

from analytic number theory to show that this conjecture is true for all graphs

having minimum degree Ω(log |V (G)|). They also proved that if G is a graph with

|V (G)| ≥ 4 vertices and maximum degree ∆(G) ≥ |V (G)|−2 then G is antimagic. It

is still an open problem to decide whether connected graphs with ∆(G) ≥ n− k and

n > n0(k) are antimagic, for any fixed k ≥ 3. In [5], it is shown that all complete

graphs, except K2, are antimagic.

It is an easy exercise to write down many antimagic labelings for most graphs,

so some further restriction on the vertex-sums is usually introduced. Bodendiek and

Walther [32] defined the concept of an (a, d)-antimagic labeling as an edge-labeling in

which the vertex-weights form an arithmetic progression starting from a and having

a common difference d. As an illustration, Figure 1.6 provides an example of a (15, 2)-

antimagic labeling for prism C5 × P2, where the vertex labels mean vertex-weights.

Bodendiek and Walther in [34] and [35] proved that the Herschel graph does


15

23

21 19

17

25

33

31 29

273

5

2

4

1

8

10

7

9

611

15

1413

12

Figure 1.6: (15, 2)-antimagic labeling of prism C5 × P2.

not admit (a, d)-antimagic labeling and obtained both positive and negative results

about (a, d)-antimagic labelings for various cases of parachutes Pm,n. In [33] and [36]

there are described (a, d)-antimagic labelings for some classes of graphs, mainly for

paths, cycles and complete graphs. The paper [6] characterizes all (a, d)-antimagic

labelings of prism Cn × P2 when n is even and shows that if n is odd the prisms

Cn × P2 are(

5n+52

, 2)-antimagic.

1.2.1 Vertex-antimagic total labelings

For a total labeling f : V (G)∪E(G) → 1, 2, . . . , p+q the associated vertex-weight

of a vertex x ∈ V (G) is

wtf (x) = f(x) +∑

y∈N(x)

f(xy).

In this section we focus on the (a, d)-vertex-antimagic total labelings. Such a la-


beling of G is a bijection f from V (G) ∪ E(G) into the integers 1, 2, . . . , p + q with

the property that the set of vertex-weights is

W = wtf (x) : x ∈ V (G) = a, a + d, . . . , a + (p− 1)d,

where a > 0 and d ≥ 0 are two fixed integers. For short, we call a vertex-antimagic

total labeling a VAT labeling. If d = 0 then the (a, d)-VAT labeling is called the

vertex-magic total (VMT) labeling.

The definition of an (a, d)-VAT labeling was introduced by Baca, Bertault, Mac-

Dougall, Miller, Simanjuntak and Slamin in [8] as a natural extension of the vertex-

magic total labeling defined by MacDougall, Miller, Slamin and Wallis [67].

An (a, d)-VAT labeling f is called super if f(V ) = f(x), x ∈ V = 1, 2, . . . , p.

That is, in a super (a, d)-VAT labeling the smallest labels are assigned to the vertices.

A graph which admits a (super) (a, d)-VAT labeling is said to be (super) (a, d)-VAT.

A super (20, 2)-VAT labeling of cycle C7 is depicted in Figure 1.7.

2

5

7

6 1

4

3

12

11

14

10

9

13

8

22

28

32

30 20

26

24

Figure 1.7: Super (20, 2)-VAT labeling of cycle C7.

In [68], it is shown that wheels, fans and friendship graphs have no VMT labelings

except for the certain range of number of vertices n and for all n in this range the


VMT labelings are found. A VMT labeling for Kn, for odd n, can be found in [66],

[67] and [69], and for Kn, with n even, is given in [49] and [51]. A construction for

a VMT labeling of complete bipartite graphs Km,m is presented in [67]. In [52], it is

completely determined which complete bipartite graphs have VMT labelings. The

constructions of VMT labelings of certain regular graphs are given in [54], [65] and

[95]. The super VMT labelings for the disjoint union of m cycles of length n, mCn,

for m and n both odd, are given in [21]. Gray, MacDougall, McSorley and Wallis

[53] explore VMT labelings for a disjoint union of stars and prove that mP3∼= mK1,2

has a VMT labeling. Gomez [50] studies the super VMT labelings for the disjoint

union of regular graphs.

The basic properties of (a, d)-VAT labelings are investigated in [8] and super

(a, d)-VAT labelings are studied in [92]. In [92], it is shown how to construct the

super (a, d)-VAT labelings for certain families of graphs, including complete graphs,

complete bipartite graphs, cycles, paths and generalized Petersen graphs. In the

paper [4] there are presented some new results on existence of super (a, d)-VAT

labelings for disconnected graphs, namely a disjoint union of m copies of a regular

graph.

1.2.2 Edge-antimagic total labelings

Simanjuntak, Miller and Bertault in [87] define an (a, d)-edge-antimagic vertex label-

ing of a (p, q)-graph G = (V, E) as an injective mapping f : V (G) → 1, 2, . . . , p

such that the set of edge-weights

f(u) + f(v) : uv ∈ E(G) = a, a + d, a + 2d, . . . , a + (q − 1)d


for two non-negative integers a and d. A bijection g : V (G)∪E(G) → 1, 2, . . . , p+q

is called an (a, d)-edge-antimagic total labeling of G if the edge-weights g(u)+g(uv)+

g(v) : uv ∈ E(G) form an arithmetic sequence starting at a and having common

difference d, where a > 0 and d ≥ 0 are two fixed integers. For brevity’s sake,

we refer to an edge-antimagic vertex labeling as an EAV labeling and to an edge-

antimagic total labeling as an EAT labeling. If d = 0 then an (a, d)-EAV labeling

or an (a, d)-EAT labeling is called edge-magic vertex labeling or edge-magic total

labeling, respectively.

An (a, d)-EAT labeling is a natural extension of a notion of magic valuation

defined by Kotzig and Rosa in [64]. Figure 1.8 gives an example of a (5, 1)-EAV

labeling of cycle C7 and Figure 1.9 gives an example of a (19, 2)-EAT labeling of

cycle C8.

1

4

7

3 6

2

5

Figure 1.8: (5, 1)-EAV labeling of cycle C7.

An (a, d)-EAT labeling is called super if the smallest possible labels appear on

the vertices. A super (a, d)-EAT labeling is a natural extension of a notion of super

edge-magic total labeling defined by Enomoto, Llado, Nakamigawa and Ringel in

[43]. A graph that admits an (a, d)-EAV labeling or a super (a, d)-EAT labeling is


2

10

4

12

6

14

8

16

15

7

13

5 11

3

9

1

Figure 1.9: (19, 2)-EAT labeling of cycle C8.

called an (a, d)-EAV graph or super (a, d)-EAT graph, respectively.

In [87] Simanjuntak, Miller and Bertault studied the properties of (a, d)-EAV

labeling and (a, d)-EAT labeling and gave constructions of (a, d)-EAT labelings for

cycles and paths. Baca, Lin, Miller and Simanjuntak [7] presented some relationships

between (a, d)-EAV labeling, (a, d)-EAT labeling and other labelings, namely, edge-

magic vertex labeling and edge-magic total labeling. In the paper [10], the super

(a, d)-edge-antimagic properties of certain classes of graphs are studied, including

friendship graphs, wheels, fans, complete graphs and complete bipartite graphs.

For an (a, d)-EAT labeling, the minimum possible edge-weight is at least 1+2+3.

Consequently a ≥ 6. The maximum possible edge-weight is no more than (p + q −

2) + (p + q − 1) + (p + q) = 3p + 3q − 3.

Thus;

a + (q − 1)d ≤ 3p + 3q − 3,

d ≤ 3p + 3q − 9

q − 1. (1.1)

This gives an upper bound for the parameter d for an (a, d)-EAT labeling of G.


Next two theorems proved by Baca, Lin, Miller and Simanjuntak [7] present

relationship between (a, d)-EAT labelings and edge-magic vertex labelings.

Theorem 1. [7] If an (p, q)-graph G has an edge-magic vertex labeling then G has

an (a + p + 1, 1)-EAT labeling.

Theorem 2. [7] Let G be an (p, q)-graph which admits total labeling and whose edge

labels constitute an arithmetic progression with difference d. Then the following are

equivalent.

(i) G has an edge-magic total labeling,

(ii) G has an (a− (q − 1)d, 2d)-EAT labeling.

Assume that a (p, q)-graph G has a super (a, d)-EAT labeling f : V (G)∪E(G) →

1, 2, . . . , p + q. The minimum possible edge-weight in the labeling f is at least

1 + 2 + p + 1 = p + 4. Thus, a ≥ p + 4. On the other hand, the maximum possible

edge-weight is at most (p− 1) + p + (p + q) = 3p + q − 1.

So

a + (q − 1)d ≤ 3p + q − 1

and

d ≤ 2p + q − 5

q − 1. (1.2)

Thus, we have the upper bound for the difference d. From (1.2) it follows that

for any connected (p, q)-graph, where p− 1 ≤ q, the feasible value d is no more than

3.

The next theorem is useful for extending an edge-antimagic vertex labeling to

a super edge-antimagic total labeling.


Theorem 3. [7] If (p, q)-graph G has an (a, d)-EAV labeling then

(i) G has a super (a + p + 1, d + 1)-EAT labeling, and

(ii) G has a super (a + p + q, d− 1)-EAT labeling.

The super (a, d)-EAT labelings for generalized Petersen graph P (n, m) are given

in [9] and [78]. Sugeng, Miller, Slamin and Baca [93] described super (a, d)-EAT la-

belings for stars and caterpillars. In [94] there are studied super (a, d)-EAT labelings

for ladders, generalized prisms and generalized antiprisms.

Many authors investigated the existence of super edge-antimagic labelings for

disconnected graphs. Ivanco and Luckanicova [59] described some constructions of

super edge-magic total labelings for disconnected graphs, namely, nCk ∪ mPk and

K1,m∪K1,n. Super (a, d)-EAT labelings for Pn∪Pn+1, nP2∪Pn and nP2∪Pn+2 have

been described by Sudarsana, Ismaimuza, Baskoro and Assiyatun in [91]. Dafik,

Miller, Ryan and Baca investigated the super edge-antimagicness for the disjoint

union of cycles mCn and for disjoint union of paths mPn in [41] and for a variety

of disjoint unions of caterpillars in [13]. In the paper [40] they investigated the

existence of antimagic labelings of disjoint unions of s-partite graphs. Baca, Lin and

Muntaner-Batle provide super (a, d)-EAT labelings for path-like trees in [12] and for

forest in which every component is a specific kind of tree in [16].

1.3 Graceful labelings

A (p, q)-graph G = (V, E) is said to be labeled by mapping φ if to each vertex

x ∈ V (G) is assigned a non-negative integer value φ(x) and to each edge xy ∈ E(G)

is assigned the value |φ(x) − φ(y)|. The labeling φ is called graceful if φ : V →


1, 2, . . . , q + 1 is an injection and if all edges of G have assigned distinct labels

from the set 1, 2, . . . , q. A graph is called graceful if it admits a graceful labeling.

Graceful labeling was introduced by Rosa in 1967 [82]. However, Rosa called this

labeling β-valuation and used the injection φ : V → 0, 1, 2, . . . , q. Several years

later Golomb [48] studied the same type of labeling and called this labeling graceful.

Graceful labelings were introduced to attack Ringel’s conjecture [80], i.e. that the

complete graph K2n+1 is decomposable into 2n+1 subgraphs that are all isomorphic

to a given tree of size n.

Figure 1.10 illustrates a graceful labeling of a tree on 8 vertices. The weights of

the edges induced by the graceful labeling are given in italic.

5

1 7

4 6

2 38

6

4

7

3

51

2

Figure 1.10: Graceful labeling of a tree.

It is known that not every graph is graceful, for instance we can consider the

complete graph Kn, n ≥ 5, and the cycle Cn, n ≡ 1 or 2 (mod 4). The smallest

graph, in order and size, that is not graceful is C3 ∪K1,1.

The Ringel-Kotzig’s conjecture that all trees are graceful is a very popular open

problem. Among the trees known to be graceful are caterpillars [82], trees with at

most 4 end-vertices [58], trees with diameter at most 5 [57] and trees with at most

27 vertices [3].


1.3.1 α-labelings

The graceful labeling φ with the property that there exists an integer λ such that

for each edge xy either

φ(x) ≤ λ < φ(y) or φ(y) ≤ λ < φ(x),

is called an α-labeling. The number λ is called the boundary value of φ.

A graph with an α-labeling is necessarily bipartite and the boundary value must

be the smaller of two vertex labels that yield the edge label 1. A graph that admits

an α-labeling is called an α-graph.

Some methods for constructing the graceful labelings and α-labelings for certain

families of trees can be found in [2, 22, 23, 24, 25, 37, 42, 83, 88].

Example of an α-graph with the boundary value λ = 7 is depicted on Figure

1.11, where the integers on the edges are their weights.

1 10 5 8

2 3 4 6 7

12 11 9

9 5 311 10

4

78 6 2 1

Figure 1.11: α-labeling of a caterpillar.

Many results appeared about graceful labeling. A recent survey of graceful la-

beling can be found in the Gallian’s comprehensive dynamic survey [46]. In this

thesis, we study the properties of edge-antimagic labelings. We use the connection

between graceful labelings and edge-antimagic labelings of trees for generating large


classes of edge-antimagic total trees from smaller graceful trees. These results will

be presented in Chapter 4.

Chapter 2

Super (a, d)-EAT labelings for

d even

Although many results on (a, d)-EAT labelings have already been published (see

[46]), there are still many problems that we can try to solve. In this chapter we

will mainly investigate the existence of super (a, d)-EAT labelings for disconnected

graphs. We will concentrate on the following problem: If a graph G is (super) (a, 2)-

EAT, is the disjoint union of m copies of the graph G (denoted by mG) (super)

(a, 2)-EAT as well? Additionally, we will present some necessary conditions for the

existence of (a, d)-EAT labelings for d even.

Figueroa-Centeno, Ichishima and Muntaner-Batle in [44] showed that a graph

G admits a super edge-magic total labeling if and only if G admits a (a, 1)-EAV

labeling.

Lemma 1. [44] A (p, q)-graph G is super edge-magic total if and only if there exists

a bijective function f : V (G) → 1, 2, . . . , p, such that the set

S = f(u) + f(v) : uv ∈ E(G)

21

Chapter 2. Super (a, d)-edge-antimagic total labelings for d even 22

consists of q consecutive integers. In such a case, f can be extended to a super

edge-magic total labeling of G with valence k = p + q + s, where s = min(S) and

S = k − (p + 1), k − (p + 2), . . . , k − (p + q).

Theorem 3 allows us to extend the previous known results on super edge-magic

total labelings onto super (a − q + 1, 2)-EAT labelings. However, the condition in

Theorem 3 is only sufficient for the existence of a super (a, 2)-EAT labeling from

the existence of a super edge-magic total labeling of a graph. For example, let us

consider two copies of a path on 3 vertices. In [45] it is proved that 2P3 is not

super edge-magic total, but it is super (a, 2)-EAT, see Figure 2.1. Note, that 2P3 is

edge-magic total, see Figure 2.2.

2 3 5 6

1 4

9 10 7 8

Figure 2.1: Super (12, 2)-EAT labeling of 2P3.

1 2 3 5

9 4

7 6 10 8

Figure 2.2: Edge-magic total labeling of 2P3.

In [45] Figueroa-Centeno, Ichishima and Muntaner-Batle proved


Theorem 4. [45] If G is a (super) edge-magic total bipartite or tripartite graph and

m is odd, then mG is (super) edge-magic total.

The next corollary immediately follows from the previous theorem.

Corollary 1. If T is a (super) edge-magic total tree and m is odd, then mT is

(super) edge-magic total.

Kotzig and Rosa [64] have shown that all cycles are edge-magic total. Thus, we

have the following corollary.

Corollary 2. If m is odd and n > 1, then the 2-regular graph mC2n is edge-magic

total.

Baca, Lin and Muntaner-Batle [12] proved that every path on n vertices has

a super edge-magic labeling. From Theorem 4, it follows

Corollary 3. If m is odd, m ≥ 3 and n ≥ 2, then the graph mPn is super edge-magic.

2.1 Super (a, 2)-edge-antimagic total labelings for

the disjoint union of graphs

From Theorem 4, it follows that if G is a super edge-magic total tripartite graph

and m is odd, then mG is super (a, 2)-EAT. We are able to extend this result by the

following theorem.

Theorem 5. [18] If G is a (super) (a, 2)-EAT tripartite graph and m is odd, then

mG is (super) (a′, 2)-EAT.


Proof. Let G be a (super) (a, 2)-EAT tripartite (p, q)-graph with the partite sets

V1, V2 and V3. Then E(G) = V1V2 ∪ V2V3 ∪ V1V3, where the juxtaposition of two

partite sets denotes the set of edges between those two sets.

Let f : V (G) ∪E(G) −→ 1, 2, . . . , p + q be a (super) (a, 2)-EAT labeling of G.

By the symbol xi we denote the element (a vertex or an edge) in the ith copy of

mG corresponding to the element x ∈ V (G) ∪ E(G).

We define a new labeling g of mG, for m odd, in the following way.

g(xi) =

m[f(x)− 1] + i if x ∈ V1 ∪ V2V3,

m[f(x)− 1] + i + m+12

if x ∈ V2 ∪ V1V3 and i < m2,

m[f(x)− 1] + i− m+12

+ 1 if x ∈ V2 ∪ V1V3 and i > m2,

m[f(x)− 1] + 2i if x ∈ V3 ∪ V1V2 and i < m2,

m[f(x)− 1] + 2i−m if x ∈ V3 ∪ V1V2 and i > m2.

Let t ∈ 1, 2, . . . , p + q. We consider the following three cases:

Case 1. If the number t is assigned by the labeling f to some element of V1∪V2V3,

then the corresponding elements in the copies of G in mG have labels

m(t− 1) + 1 in G1

m(t− 1) + 2 in G2

......

m(t− 1) + i in Gi

m(t− 1) + i + 1 in Gi+1

......

mt− 1 in Gm−1

mt in Gm


i.e. the numbers m(t− 1) + 1, m(t− 1) + 2, . . . ,mt.


then the corresponding elements in the copies of G in mG receive labels

mt + 3−m2

in G1

mt + 5−m2

in G2

......

mt in Gm−12

m(t− 1) + 1 in Gm+12

m(t− 1) + 2 in Gm+32

......

mt + 1−m2

in Gm

thus, the numbers m(t− 1) + 1, m(t− 1) + 2, . . . ,mt.


then the corresponding elements in the copies of G in mG receive labels

m(t− 1) + 2 in G1

m(t− 1) + 4 in G2

......

mt− 1 in Gm−12

m(t− 1) + 1 in Gm+12

m(t− 1) + 3 in Gm+32

......

mt in Gm

hence the corresponding labels are m(t− 1) + 1, m(t− 1) + 2, . . . ,mt.


Thus, the set of the labels in mG, corresponding to the value t, is independent

on the labeled element. It means that the labeling g is evidently total and assigns

the numbers 1, 2, . . . ,m(p + q) to the elements of mG. Moreover, if the labeling f

is super, then the smallest possible labels are also used to label the vertices in mG

and thus g is also super.

In the next part, we will calculate the edge-weight of an edge uv ∈ E(Gi). We

will again distinguish three cases. If u ∈ V i1 and v ∈ V i

2 , if u ∈ V i1 and v ∈ V i

3 and if

u ∈ V i2 and v ∈ V i

3 . By the symbol V ij , j = 1, 2, 3 and i = 1, 2, . . . ,m, we denote the

vertex set corresponding to the vertex set Vj in the ith copy of G.

It is easy to verify that in all cases we obtain for the edge-weights:

g(ui) + g(vi) + g(uivi)

=

m[f(u) + f(v) + f(uv)− 3] + m+1

2+ 4i if i < m

2,

m[f(u) + f(v) + f(uv)− 3] + m+12− 2m + 4i if i > m

2.

Thus, the edge-weight A of some edge uv in G, A = f(u)+f(v)+f(uv), corresponds

to the following edge-weights in mG:

m(A− 3) + m+12

+ 4 in G1

m(A− 3) + m+12

+ 8 in G2

......

m(A− 3) + m+12

+ 2m− 2 in Gm−12

m(A− 3) + m+12

+ 2 in Gm+12

m(A− 3) + m+12

+ 6 in Gm+32

......

m(A− 3) + m+12

+ 2m in Gm.


It means that the edge-weights are:

m(A− 3) +m + 1

2+ 2, m(A− 3) +

m + 1

2+ 4, . . . ,m(A− 1) +

m + 1

2.

As f is (a, 2)-EAT labeling then the edge-weights in G are:

a, a + 2, a + 4, . . . , a + 2(q − 1).

Thus, to the edge-weight A + 2 in g the corresponding edge-weights in mG are:

m(A− 1) +m + 1

2+ 2, m(A− 1) +

m + 1

2+ 4, . . . ,m(A + 1) +

m + 1

2,

hence, the edge-weights in mG again form an arithmetic sequence with the difference

2 and the initial term m(a− 3) + m+12

+ 2. This concludes the proof.

Immediately, from the previous theorem, we get the following result:

Corollary 4. [18] If G is a (super) (a, 2)-EAT bipartite graph and m is odd, then

mG is (super) (a′, 2)-EAT.

In [59], Ivanco and Luckanicova proved a more general result than the one in

Theorem 4 for a disjoint union of edge-magic total graphs. A mapping c : V (G) ∪

E(G) −→ 1, 2, 3 is called an e-m-coloring of a graph G if c(u), c(v), c(uv) =

1, 2, 3 for any edge uv of G. Ivanco and Luckanicova proved:

Theorem 6. [59] Let m be an odd positive integer. For i = 1, 2, . . . ,m, let Gi, gi

and ci be an edge-magic total graph with pi vertices and qi edges, an edge-magic total

labeling of Gi with its magic number σi and an e-m-coloring of Gi, respectively.

Suppose that the following conditions are satisfied:

(1) there is an integer σ such that σi = σ for all i = 1, 2, . . . ,m,


(2) if gi(x) = gj(y), then ci(x) = cj(y) for all i, j = 1, 2, . . . ,m, x ∈ V (Gi)∪E(Gi)

and y ∈ V (Gj) ∪ E(Gj),

(3) there is an integer r such that r = p1 + q1 ≥ · · · ≥ pm + qm ≥ r − 1.

Then, the disjoint union ∪mi=1Gi is an edge-magic total graph.

Moreover, if all gi are super edge-magic total labelings and p1 = p2 = · · · = pm,

then ∪mi=1Gi is a super edge-magic total graph.

We present a similar result for (super) (a, 2)-EAT graphs.

Theorem 7. [18] Let m be an odd positive integer. For i = 1, 2, . . . ,m, let Gi, fi

and ci be an (a, 2)-EAT graph with pi vertices and qi edges, an (a, 2)-EAT labeling

of Gi and an e-m-coloring of Gi, respectively.

Suppose that the following conditions are satisfied:

(1) if fi(x) = fj(y), then ci(x) = cj(y) for all i, j = 1, 2, . . . ,m, x ∈ V (Gi)∪E(Gi)

and y ∈ V (Gj) ∪ E(Gj),

(2) there is an integer r such that r = p1 + q1 ≥ · · · ≥ pm + qm ≥ r − 1.

Then, the disjoint union ∪mi=1Gi is an (a′, 2)-EAT graph.

Moreover, if all fi are super (a, 2)-EAT labelings and p1 = p2 = · · · = pm, then

∪mi=1Gi is a super (a′, 2)-EAT graph.

Proof. For i = 1, 2, . . . ,m, let Gi, fi and ci be an (a, 2)-EAT graph with pi vertices

and qi edges, an (a, 2)-EAT labeling of Gi and an e-m-coloring of Gi, respectively.


We define a new labeling g of ∪mi=1Gi, for m odd, in the following way.

g(xi) =

m[fi(x)− 1] + i if ci(x) = 1,

m[fi(x)− 1] + i + m+12

if ci(x) = 2 and i < m2,

m[fi(x)− 1] + i− m+12

+ 1 if ci(x) = 2 and i > m2,

m[fi(x)− 1] + 2i if ci(x) = 3 and i < m2,

m[fi(x)− 1] + 2i−m if ci(x) = 3 and i > m2.

According to condition (1), it is not difficult to check that the labeling g uses

each integer 1, 2, . . . , |V (∪mi=1Gi) ∪ E(∪m

i=1Gi)| exactly once. As fi is an (a, 2)-EAT

labeling of Gi and ci is an e-m-coloring of Gi, then analogously as in the proof of

Theorem 5 we show, that ∪mi=1Gi is an (a′, 2)-EAT graph. Moreover, if all fi are

super (a, 2)-EAT labelings and p1 = p2 = · · · = pm, then

1 ≤ g(u) ≤ (pi − 1)m + m = |V (∪mi=1Gi)|.

Thus, g is a super (a′, 2)-EAT labeling, too.

2.2 Conditions for non-existence of (super) (a, d)-

edge-antimagic total labelings

In the literature, there are some known conditions for the non-existence of the (a, d)-

EAT labelings for some graphs depending on the order and the size of a graph. In

[87] it is proved

Theorem 8. [87] A graph with all vertices of odd degrees cannot have an (a, d)-EAT

labeling with a and d both even.


Theorem 9. [87] Let G be a (p, q)-graph with all vertices of odd degrees. If q ≡ 0

(mod 4) and p ≡ 2 (mod 4) then G has no (a, d)-EAT labeling.

Theorem 10. [87] Suppose G is a (p, q)-graph whose every vertex has an odd degree.

Then in the following cases G has no (a, d)-EAT labeling.

(i) q ≡ 1 (mod 4), p ≡ 0 (mod 4), and a even,

(ii) q ≡ 1 (mod 4), p ≡ 2 (mod 4), and a odd,

(iii) q ≡ 2 (mod 4), p ≡ 2 (mod 4), and d odd,

(iv) q ≡ 3 (mod 4), p ≡ 0 (mod 4), a even, and d odd.

Moreover, for edge-magic total graphs, the following is proved in [81]

Theorem 11. [81] Let G be a (p, q)-graph with all vertices of odd degrees. If q ≡ 0

(mod 2) and p + q ≡ 2 (mod 4) then G has no edge-magic total labeling.

Baca and Youssef, in [11], used parity arguments to find the conditions on p, q

and d, for which a graph with p vertices and q edges cannot have a super (a, d)-EAT

labeling.

Theorem 12. [11] Let G be a regular (p, q)-graph of odd degree. If p ≡ 2 (mod 4)

and q ≡ 0 (mod 2) then G is not super (a, d)-EAT, for every odd d.

Theorem 13. [11] Let G be a regular (p, q)-graph of even degree. If q ≡ 2 (mod 4)

then G is not super (a, d)-EAT, for every even d.

These results are based on the arguments using divisibility. More precisely, we

get the following lemma:


Lemma 2. [18] Let G be a (p, q)-graph with all vertices of odd degrees and let d be

an even integer. If one of the following conditions holds

(i) q ≡ 0 (mod 4) and p ≡ 1 (mod 4) or p ≡ 2 (mod 4),

(ii) q ≡ 2 (mod 4) and p ≡ 0 (mod 4) or p ≡ 3 (mod 4)

then G has no (a, d)-EAT labeling.

Proof. Assume that (p, q)-graph G has an (a, d)-EAT labeling f . The sum of all the

edge-weights is

∑xy∈E(G)

w(xy) =

q−1∑i=0

(a + id) = aq +q(q − 1)d

2. (2.1)

In the computation of the edge-weights of G, each edge label is used once and the

label of vertex x is used d(x) times. The sum of all vertex labels and edge labels

used to calculate the edge-weights is thus, equal to:

∑xy∈E(G)

f(xy) +∑

x∈V (G)

d(x)f(x) =(p + q)(p + q + 1)

2+

∑x∈V (G)

(d(x)− 1)f(x). (2.2)

Combining (2.1) and (2.2) gives:

aq +q(q − 1)d

2=

(p + q)(p + q + 1)

2+

∑x∈V (G)

(d(x)− 1)f(x). (2.3)

If G is a graph with all vertices of odd degree, d is even and if one of the conditions

in the lemma holds, then using the parity considerations on the left hand and on the

right hand side of the formula we get a contradiction.

For an even number of copies of a graph we have:


Theorem 14. [18] Let d, k be positive integers, d ≡ 0 (mod 2), k ≡ 1 (mod 2). Let

G be a (p, q)-graph with all vertices of odd degrees. If the size and the order of G

have a different parity then the graph 2kG has no (a, d)-EAT labeling.

Proof. Consider a graph 2kG, where G is a graph with all vertices of odd degrees.

Let k ≡ 1 (mod 2).

If the size is odd and the order is even then:

|E(2kG)| = 2kq ≡ 2 (mod 4), |V (2kG)| = 2kp ≡ 0 (mod 4).

If the size is even and the order is odd then:

|E(2kG)| = 2kq ≡ 0 (mod 4), |V (2kG)| = 2kp ≡ 2 (mod 4).

Thus, according to Lemma 2, the graph 2kG is not (a, d)-EAT for d even.

For example, let us consider a star K1,n. In [93] it is proved that every star

is super edge-magic total and super (a, 2)-EAT. As the star is a bipartite graph,

then the odd number of copies of a star K1,n is super edge-magic total and super

(a, 2)-EAT according to Theorem 4 and Theorem 5. However, for n ≡ 1 (mod 2)

using Theorem 14 we get that (4k + 2) copies of K1,n is neither edge-magic total nor

(a, 2)-EAT. Thus also the graph G is (a, d)-EAT for d ≡ 0 (mod 2), in many cases

there does not exist such a labeling of even number of copies of G. This indicates

that there exists no general construction of (a, d)-EAT labeling for even number of

copies of a graph for d even.

Chapter 3

Super (a, d)-EAT labelings for

d odd

The (a, d)-EAT and super (a, d)-EAT labelings are two among several other “magic-

type” labelings, that have been introduced. Often results on one or more magic-type

labelings can be adapted or combined to obtain results on a different type. This

idea has been studied by Figueroa-Centeno, Ichishima, and Muntaner-Batle [44]. In

this chapter we study a set of problems which are similar to the problems studied

in [65] for vertex-magic total labelings. We prove that every even regular graph and

every odd regular graph with a 1-factor are super (a, 1)-EAT. We also introduce some

constructions of non-regular super (a, 1)-EAT graphs. For an exhaustive survey on

various magic-type labelings we again recommend [46].

33

Chapter 3. Super (a, d)-edge-antimagic total labelings for d odd 34


regular graphs

Results in this section are based on the following Petersen Theorem.

Theorem 15. (Petersen Theorem) Let G be a 2r-regular graph. Then there exists

a 2-factor in G.

Notice that after removing edges of the 2-factor guaranteed by the Petersen The-

orem we have again an even regular graph. Thus, by induction, an even regular

graph has a 2-factorization.

The construction in the following theorem allows us to find a super (a, 1)-EAT

labeling of any even regular graph. Notice that the construction does not require the

graph to be connected.

Theorem 16. [19] Let G be a graph on p vertices that can be decomposed into two

factors G1 and G2. If G1 is edge-empty or if G1 is a super (2p + 2, 1)-EAT graph

and G2 is a 2r-regular graph then G is super (2p + 2, 1)-EAT.

Proof. First we start with the case when G1 is not edge-empty. Since G1 is a super

(2p + 2, 1)-EAT graph with p vertices and q edges, there exists a total labeling

f : V (G1) ∪ E(G1) → 1, 2, . . . , p + q such that

f(v) + f(uv) + f(v) : uv ∈ E(G) = 2p + 2, 2p + 3, . . . , 2p + q + 1.

By the Petersen Theorem there exists a 2-factorization of G2. We denote the 2-

factors by Fj, j = 1, 2, . . . , r. Without loss of generality we can suppose that

V (G) = V (G1) = V (Fj) for all j and E(G) = ∪rj=1E(Fj) ∪ E(G1). Each factor

Fj is a collection of cycles. We order and orient the cycles arbitrarily. Now by the


symbol eoutj (vi) we denote the unique outgoing arc from the vertex vi in the factor

Fj.

We define a total labeling g of G in the following way:

g(v) = f(v) for v ∈ V (G),

g(e) =

f(e) for e ∈ E(G1),

q + (j + 1)p + 1− f(vi) for e = eoutj (vi).

The vertices are labeled by the first p integers, the edges of G1 by the next q

labels and the edges of G2 by consecutive integers starting at p + q + 1. Thus g is

a bijection V (G) ∪ E(G) → 1, 2, . . . , p + q + pr since |E(G)| = q + pr.

It is not difficult to verify, that g is a super (2p + 2, 1)-EAT labeling of G. The

weights of the edges e in E(G1) are wg(e) = wf (e). The weights form the progression

2p + 2, 2p + 3, . . . , 2p + q + 1. For convenience we denote by vk the unique vertex

such that vivk = eoutj (vi) in Fj. The weights of the edges in Fj, j = 1, 2, . . . , r are

wg(eoutj (vi)) = wg(vivk) = g(vi) + (q + (j + 1)p + 1− f(vi)) + g(vk)

= f(vi) + q + (j + 1)p + 1− f(vi) + f(vk) = q + (j + 1)p + 1 + f(vk)

for all i = 1, 2, . . . , p and j = 1, 2, . . . , r. Since Fj is a factor, the set f(vk) : vk ∈

Fj = 1, 2, . . . , p. Hence, we have that the set of the edge-weights in the factor Fj

is q + (j + 1)p + 2, q + (j + 1)p + 3, . . . , q + (j + 1)p + p + 1 and thus the set of all

edge-weights in G is 2p + 2, 2p + 3, . . . , q + (r + 2)p + 1.

If G1 is edge-empty it is enough to take q = 0 and proceed with the labeling of

factors Fj.

By taking an edge-empty graph G1 we have the following theorem:


Theorem 17. [19] All even regular graphs of order p with at least one edge are super

(2p + 2, 1)-EAT.

Baca, Lin and Semanicova-Fenovcıkova [15] proved the following theorem:

Theorem 18. [15] Let Gi be a super (a, 1)-EAT graph of order p and size q, i =

1, 2, . . . ,m. Then the disjoint union⋃m

i=1 Gi is also a super (b, 1)-EAT graph.

Following from Theorem 18 we get;

Corollary 5. [15] Let G be a super (a, 1)-EAT graph. The disjoint union of arbitrary

number of copies of G, i.e. mG, m ≥ 1, also admits a super (b, 1)-EAT labeling.

Thus we have the following theorem (we prefer to call it a theorem though it is

just a corollary of Theorem 17 and Theorem 18).

Theorem 19. [19] Let Gi be an 2r-regular graph of order p, i = 1, 2, . . . ,m. Then

the disjoint union⋃m

i=1 Gi admits a super (a, 1)-EAT labeling.

The construction from Theorem 16 can be extended also to the case when G1 is

not a factor. One can add isolated vertices to a graph and keep the property of being

super (a, 1)-EAT. A graph consisting of m isolated vertices is denoted by mK1. We

can obtain the following lemma.

Lemma 3. [19] If G is a super (a, 1)-EAT graph then also G ∪ mK1 is a super

(a + m + 2t, 1)-EAT graph for all t ∈ 0, 1, . . . ,m.

Proof. Since G is a super (a, 1)-EAT graph with p vertices and q edges, there exists

such a total labeling f : V (G) ∪ E(G) → 1, 2, . . . , p + q that

f(v) + f(uv) + f(v) : uv ∈ E(G) = a, a + 1, . . . , a + q − 1.


Let t be any fixed integer from 0, 1, . . . ,m. Let (c1, c2, . . . , cm) be any permutation

of the integers in 1, 2, . . . , p + m \ t + 1, t + 2, . . . , t + p. We denote the vertices

of mK1 by vc1 , vc2 , . . . , vcm arbitrarily. Now we define a labeling g of the graph

H = G ∪mK1.

g(v) =

f(v) + t for v ∈ V (G),

i for v = vi, where vi ∈ mK1,

g(e) = f(e) + m for e ∈ E(H).

Obviously g is a bijection V (H) ∪ E(H) → 1, 2, . . . , p + q + m. The edges are

labeled by the q highest labels and the vertices by the first p + m integers. It is

easy to verify that g is a super (a + m + 2t, 1)-EAT labeling of H, since any edge

uv ∈ E(H) is also in E(G).

wg(uv) = g(u) + g(uv) + g(v)

= (f(u) + t) + (f(uv) + m) + (f(v) + t) = wf (uv) + m + 2t

and the claim follows.

Notice that we can find m + 1 different (up to isomorphism) super (b, 1)-EAT

labelings of G ∪mK1 but all with the same parity of the smallest edge-weight.

Next we show that also all odd regular graphs with a perfect matching are super

(a, 1)-EAT.

Lemma 4. [19] Let k, m be positive integers. Then the graph kP2 ∪mK1 is super

(2(2k + m) + 2, 1)-EAT.

Proof. We denote the vertices of the graph G ∼= kP2∪mK1 by the symbols v1, v2, . . . ,

v2k+m in such a way that E(G) = vivk+m+i : i = 1, 2, . . . , k and the remaining

vertices are denoted arbitrarily by the unused symbols.


We define the labeling f : V (G) ∪ E(G) → 1, 2, . . . , 3k + m in the following

way;

f(vj) = j for j = 1, 2, . . . , 2k + m,

f(vivk+m+i) = 3k + m + 1− i for i = 1, 2, . . . , k.

It is easy to see that f is a bijection and that the vertices of G are labeled by the

smallest possible numbers. For the edge-weights we get;

wf (vivk+m+i) =f(vi) + f(vivk+m+i) + f(vk+m+i)

=i + (3k + m + 1− i) + (k + m + i)

=2(2k + m) + 1 + i for i = 1, 2, . . . , k.

Thus f is a super (2(2k + m) + 2, 1)-EAT labeling of G.

Now by taking m = 0 and observing that the number of vertices in kP2 is 2k, we

immediately obtain the following theorem:

Theorem 20. [19] If G is an odd regular graph on p vertices that has a 1-factor,

then G is super (2p + 2, 1)-EAT.

Unfortunately the construction does not solve the existence of (a, 1)-EAT label-

ings for all odd regular graphs, it only works for those that contain a 1-factor. We

know that some graphs that arose by Cartesian products also satisfy this property,

therefore, we can obtain the following corollary:

Corollary 6. [19] Let G be a regular graph. Then the Cartesian product G×K2 is

a super (a, 1)-EAT graph.


Proof. If G is a (2r+1)-regular graph then the product G×K2 is (2r+2)-regular and

by Theorem 17 it is super (a, 1)-EAT. If G is 2r-regular then G×K2 is a (2r + 1)-

regular graph with a 1-factor and thus, according to Theorem 20, is super (a, 1)-

EAT.

Let us point out that many results published on super (a, 1)-EAT labelings (see

[46]) follow from Theorems 17 and 20 as a corollary.


non-regular graphs

Theorem 16 is not restricted to regular graphs, it can be used also to obtain super

(a, 1)-EAT labelings of certain non-regular graphs. We illustrate the technique on

a couple of examples. First we introduce the following lemmas.

Lemma 5. [19] Let k, m be positive integers, k < 2m+3. Then the graph K1,k∪mK1

is super (2(k + m + 1) + 2, 1)-EAT.

Proof. We distinguish two subcases according to the parity of k.

Let k be an odd positive integer. We denote the vertices of the graph G ∼= K1,k ∪

mK1 by the symbols v1, v2, . . . , vk+m+1 in such a way that E(G) = vivm+2+ k−12

: i =

1, 2, . . . , k and the remaining vertices are denoted arbitrarily by the unused symbols.

Notice that it is possible to use such notation as k < 2m + 3.

We define the labeling f : V (G)∪E(G) → 1, 2, . . . , 2k +m+1 in the following


way:

f(vj) = j for j = 1, 2, . . . , k + m + 1,

f(vivm+2+ k−12

) =

m + 3k+1

2+ i for i = 1, 2, . . . , k+1

2,

m + k+12

+ i for i = k+32

, k+52

, . . . , k.

For the edge-weights we have;

wf (vivm+2+ k−12

) =f(vi) + f(vivm+2+ k−12

) + f(vm+2+ k−12

)

=

i +(m + 3k+1

2+ i

)+

(m + 2 + k−1

2

)= 2m + 2k + 2 + 2i for i = 1, 2, . . . , k+1

2,

i +(m + k+1

2+ i

)+

(m + 2 + k−1

2

)= 2m + k + 2 + 2i for i = k+3

2, k+5

2, . . . , k,

i.e. the set of the edge-weights is 2m+2k +4, 2m+2k +5, . . . , 2m+3k +3. Thus,

for 2m+3 > k, k ≡ 1 (mod 2), f is a super (2(k +m+1)+2, 1)-EAT labeling of G.

Notice that the edge v k+12

vm+2+ k−12

is labeled under the labeling f by the highest

label m + 2k + 1 and has also the maximal edge-weight 2m + 3k + 3. Thus, it is

possible to delete this edge from G and the obtained graph K1,(k−1) ∪ (m+1)K1 will

also be super (2(k +m+1)+2, 1)-EAT. It means that it is also possible to construct

the required labeling in the case when the star has even number of pending edges

(for k even).

Lemma 6. [19] Let k, m be positive integers, let m be even. Let H be an arbitrary

2-regular graph of order k. Then the graph H ∪mK1 is super (2(k +m)+2, 1)-EAT.

Proof. According to Theorem 16 the graph H is super (2k + 2, 1)-EAT. Using Lem-

ma 3, for t = m2, we get that H ∪mK1 is a super (2(k + m) + 2, 1)-EAT graph.


Lemma 7. [19] Let k, m be positive integers, let m be even. Then the graph Pk∪mK1

is super (2(k + m) + 2, 1)-EAT.

Proof. It is known that the path on k vertices is super (2k + 2, 1)-EAT, see [7] or

[12]. According to Lemma 3, for t = m2, we get that the graph Pk ∪ mK1 is super

(2(k + m) + 2, 1)-EAT.

Immediately, from the previous lemmas and Theorem 16, we can see that it is

possible to “add” certain edges to an even regular graph and obtain a super (a, 1)-

EAT graph. The edges are added in such a way that the graph induced by these

edges is isomorphic to a collection of independent edges, to a star, to a 2-regular

graph, or to a path.

Theorem 21. [19] Let k, m be positive integers. Let G be a graph on p vertices that

can be decomposed into two factors G1 and G2. If G2 is a 2r-regular graph and either

(1) G1 is the graph kP2 ∪mK1, or

(2) G1 is the graph K1,k ∪mK1 for k < 2m + 3, or

(3) H is an arbitrary 2-regular graph of order k and G1∼= H ∪mK1 for even m,

or

(4) G1 is the graph Pk ∪mK1 for even m,

then the graph G is super (2p + 2, 1)-EAT.

Proof. Since the smallest edge-weight in G1 in case (1) is 2(2k+m)+2 = 2p+2 then

the claim immediately follows by Lemma 4 and Theorem 16. By a similar argument

one can prove cases (2), (3), and (4) using Theorem 16 and Lemmas 5, 6, and 7,

respectively.


Notice that in Lemmas 4, 5, 6 and 7 by taking m = 0 we obtain an (2p′+2, 1)-EAT

labeling of the corresponding graph on p′ = p−m vertices.

Now adding m isolated vertices one can obtain by Lemma 3 not one, but m + 1

different super (a, 1)-EAT labelings of the graph G1 in each of the cases of Theo-

rem 21. This again implies several different super (2p + 2, 1)-EAT labelings of the

graph G in Theorem 21. There can be significantly more than m + 1 different label-

ings, since we may choose various orderings of an orientation of the 2-factors Fj of

G2 (as described in the proof of Theorem 16).

Chapter 4

Generating large classes of EAT

trees

In this chapter we will describe a construction to obtain an α-tree from smaller

graceful trees. We will use the connection between α-labeling as a special class of

graceful labelings and edge-antimagic vertex labelings for generating large classes of

super (a, d)-EAT trees from smaller graceful trees.

No general method is currently known to allow one to take a tree known to

be graceful and to extend a path from it in an arbitrary position, or to identify

an arbitrary vertex with another general tree known to be graceful, in order to

produce a larger, graceful tree.

There exist various operations that generate large classes of graceful trees from

smaller graceful trees. In 1973, Stanton and Zarnke [89], were the first, developed

a nontrivial algorithm for constructing graceful trees. Their method became the

basis of many construction methods to follow.

Koh, Rogers and Tan in [61] gave a variation of Stanton and Zarnke’s construc-

43

Chapter 4. Generating large classes of edge-antimagic total trees 44

tion. This idea led to many other constructions, see [62] and [63]. We are using

one of them, described in [62], which constructs a bigger graceful tree from a given

pair of graceful trees. This construction and the connection between α-labelings and

(a, d)-EAV labelings allow us to describe a method that generates large classes of

super (a, d)-EAT trees.

4.1 Connections between α-labelings and edge-

antimagic vertex labelings for trees

From inequality (1.2), it follows that for every connected graph there is no super

(a, d)-EAT labeling with d > 4.

The following lemma gives a connection between an α-labeling of a tree and

an (a, 1)-EAV labeling. This result can be found in [75] or [17].

Lemma 8. [17] Let T be a tree of order p. If T admits an α-labeling then T also

admits an (a, 1)-EAV labeling.

In general, the converse of Lemma 8 does not hold. Figure 4.1, see [17], illustrates

(a, 1)-EAV labeling of a tree that is not an α-tree.

1 5 3

2 7

4 6

Figure 4.1: (a, 1)-EAV labeling of a tree that is not an α-tree.


With respect to Theorem 3, we can see that every α-tree also admits a super

edge-magic total and a super (a, 2)-EAT labeling.

In our next results, we establish a relationship between α-labelings and (3, 2)-

EAV labelings of trees. Notice that if a tree of size q = p − 1 is (a, 2)-EAV, then

a = 3. In fact, 1 + 2 ≤ a, and since the edge-weights are a, a + 2, . . . , a + 2(p − 2),

the inequality a + 2(p− 2) ≤ 2p− 1 holds for a ≤ 3. Therefore, a = 3.

A graph with an α-labeling is necessarily bipartite. Thus, when G is an α-graph

we denote by A, B the bipartition of its vertex set. Without loss of generality, we

may assume that |A| ≥ |B|. In [17] is shown that;

Lemma 9. [17] Let T be an α-tree. If |A| − |B| ≤ 1 then T is (3, 2)-EAV.

With respect to Theorem 3, we can see that every α-tree with |A| − |B| ≤ 1 also

admits a super (a, 1)-EAT and a super (a, 3)-EAT labeling.

Baca and Barrientos [17] proved that the converse of the previous statement also

holds. It follows from the next two lemmas.

Lemma 10. [17] Let T be a tree of order p. If |A| − |B| > 1 then there is no

(3, 2)-EAV labeling of T .

Lemma 11. [17] Let T be a tree of order p. If T does not admit an α-labeling then

neither admits an (3, 2)-EAV labeling.

From Lemmas 9, 10 and 11 it follows a connection between α-labeling of a tree

and (3, 2)-EAV labeling.

Theorem 22. [17] A tree T is (3, 2)-EAV if and only if T is an α-tree and ||A| −

|B|| ≤ 1, where A, B is the bipartition of its vertex-set.


As a consequence of Lemma 8, Theorem 22 and Theorem 3 we have the following

theorem.

Theorem 23. [17] Every α-graph of order p and size p − 1 with ||A| − |B|| ≤ 1

admits a super (a, d)-EAT labeling for all d ∈ 0, 1, 2, 3.

4.2 Construction of α-tree from smaller graceful

trees

In 1979 Koh, Rogers and Tan [62] presented a new construction of graceful trees.

They defined a new graph operation as follows.

Let G and H be two graphs and let w1, w2, . . . , wm and v1, v2, . . . , vn be their

corresponding vertex sets. Let v be an arbitrary fixed vertex in H.

Based upon the graph G, an isomorphic copy Hi of H is adjoined to each vertex wi

(i = 1, 2, . . . ,m) by identifying vi and wi, where vi is the vertex corresponding to v in

Hi. All the m copies of H just introduced are pairwise disjoint and no extra edges are

added. The graph obtained is denoted by G v©H. It is obvious that |V (G v©H)| = mn

and G v©H 6∼= H v©G and also G v©H 6∼= G u©H for v, u ∈ V (H), v 6= u in general.

Note, that Koh, Rogers and Tan used the notation G4H for this operation. The

Figure 4.3 shows a tree C6 v©K1,5 obtained by Koh, Rogers and Tan’s operation using

the cycle C6 and star K1,5 from Figure 4.2.

Theorem 24. [62] If T1 and T2 are both graceful trees then tree T1 v©T2 is also

graceful.

The next theorem uses Koh, Rogers and Tan’s construction to obtain an α-tree

from smaller graceful tree.


v

Figure 4.2: Cycle C6 and star K1,5.

Figure 4.3: Tree C6 v©K1,5.

Theorem 25. [20] Let T be a graceful tree of order n. If k is even positive integer

then the tree Pk v©T admits an α-labeling. Moreover, if T is an α-tree then Pk v©T

admits an α-labeling for every positive integer k.

Proof. Let T be a tree of order n with the bipartition A, B. Let f : V (T ) →

1, 2, . . . , n be a graceful labeling of T with the edge-weights

wf (uv) = |f(u)− f(v)| : uv ∈ E(T ) = 1, 2, . . . , n− 1.

Let x be an arbitrary fixed vertex in T . Without loss of generality, we may assume


that x ∈ A.

Now, consider k trees T 1, T 2, . . . , T k, each isomorphic to the tree T , where

Ai, Bi is the bipartition of vertex set of T i, for i = 1, 2, . . . , k, such that Ai corre-

sponds to A and Bi corresponds to B for i = 1, 2, . . . , k.

We denote the vertices of the path Pk in such a way that Pk = w1w2 . . . wk. Thus,

according to the definition of a graph Pk x©T , the vertex xi ∈ Ai is identified with

the vertex wi, for i = 1, 2, . . . , k.

For k ≡ 0 (mod 2) we define a new labeling g as follows:

g(vi) =

f(v) +

(k − i+1

2

)n for v ∈ Ai ∪Bi+1 and i = 1, 3, . . . , k − 1,

f(v) +(

i2− 1

)n for v ∈ Ai ∪Bi−1 and i = 2, 4, . . . , k.

It is easy to see that g is a bijection from V (Pk x©T ) onto 1, 2, . . . , kn and

g(v) : v ∈ Ai ∪Bi−1, i = 2, 4, . . . , k = 1, 2, . . . , kn

2

and

g(v) : v ∈ Ai ∪Bi+1, i = 1, 3, . . . , k − 1 = kn

2+ 1,

kn

2+ 2, . . . , kn.

Thus the vertex with boundary value λ = kn2

belongs to (Ak ∪ Bk−1). To see that

g is α-labeling it is enough to show that the edge-weights have distinct labels from

the set 1, 2, . . . , kn− 1. We consider two cases.

Case 1. If e = xixi+1, i = 1, 2, . . . , k − 1, then it holds

wg(e) =wg(xixi+1) = |g(xi)− g(xi+1)|

=

|[f(x) + (k − i+12

)n]− [f(x)+( i+12− 1)n]| = (k − i)n

for i = 1, 3, . . . , k − 1,

|[f(x) + ( i2− 1)n]− [f(x) + (k − i+2

2)n]| = (k − i)n

for i = 2, 4, . . . , k − 2.


Case 2. If e = uivi ∈ E(T i), without loss of generality we can suppose that

ui ∈ Ai ∪Bi+1 and vi ∈ Bi ∪ Ai+1, where i = 1, 3, . . . , k − 1. Then

wg(e) = wg(uivi) =|g(ui)− g(vi)|

=|[f(u) + (k − i+12

)n]− [f(v) + ( i+12− 1)n]|

=|(k − i)n + f(u)− f(v)|.

As f is graceful, i.e. 1 ≤ |f(u)− f(v)| ≤ n− 1, then for f(u) > f(v) we have

(k− i)n + 1 ≤ |(k− i)n + f(u)− f(v)| = (k− i)n + (f(u)− f(v)) ≤ (k− i + 1)n− 1

and for f(u) < f(v) we get

(k− i− 1)n + 1 ≤ |(k− i)n + f(u)− f(v)| = (k− i)n− (f(u)− f(v)) ≤ (k− i)n− 1.

Thus, combining the previous result we get that g is an α-labeling of Pk x©T .

Now let us consider that f is an α-labeling of T . According to the above proved,

for k even Pk x©T is an α-tree. For k odd we define a new labeling h in the following

way:

h(vi) =

f(v) + k−12

n for v ∈ A1 ∪B1 and i = 1,

f(v) + k+i−12

n for v ∈ Ai ∪Bi+1 and i = 2, 4, . . . , k − 1,

f(v) + k−i2

n for v ∈ Ai ∪Bi−1 and i = 3, 5, . . . , k.

It is easy to see that for the vertex labels it holds;

h(v) : v ∈ Ai ∪Bi−1, i = 3, 5, . . . , k

=

1, 2, . . . ,

(k − 1)n

2

,

h(v) : v ∈ A1 ∪B1

=

(k − 1)n

2+ 1,

(k − 1)n

2+ 2, . . . ,

(k + 1)n

2

,

h(v) : v ∈ Ai ∪Bi+1, i = 2, 4, . . . , k − 1

=

(k + 1)n

2+ 1,

(k + 1)n

2+ 2, . . . , kn

.


Thus, h is a bijection from V (Pk x©T ) onto 1, 2, . . . , kn. Moreover, as f is an α-

labeling with the boundary value λ, the boundary value of h is λ + (k−1)n2

.

Analogously, we can show that h is an α-labeling by proving that the set of

edge-weights is 1, 2, . . . , kn− 1.

4.3 Generating edge-antimagic trees

In this section we will discuss a method for generating large classes of super (a, d)-

EAT trees from smaller graceful trees.

The next theorems present the methods for generating the tree of type Pk v©T

with a super (a, d)-EAT labeling for a positive even integer k.

Theorem 26. [20] Let T be a graceful tree and let v be an arbitrary fixed vertex in

T . If k is an even positive integer then the tree Pk v©T admits a super (a, d)-EAT

labeling for all d ∈ 0, 1, 2, 3.

Proof. Let T be a graceful tree and v be an arbitrary fixed vertex in T . It follows

from the first part of Theorem 25 that if k is an even positive integer then the tree

Pk v©T admits an α-labeling.

If A, B is a bipartition of the vertex set of Pk v©T then it is not difficult to see

that |A| = |B| and, according to Theorem 23, we have that Pk v©T admits a super

(a, d)-EAT labeling for all d ∈ 0, 1, 2, 3.

Theorem 27. [20] Let T be an α-tree and let ||A1| − |B1|| ≤ 1, A1, B1 be the

bipartition of the vertex set of T . Let v be an arbitrary fixed vertex in T . Then for

every positive integer k the tree Pk v©T admits a super (a, d)-EAT labeling for all

d ∈ 0, 1, 2, 3.


Proof. Let T be an α-tree and v be an arbitrary fixed vertex in T . From the second

part of Theorem 25 we get that for every positive integer k the tree Pk v©T is also

the α-tree.

Moreover, if A1, B1 is the bipartition of the vertex set of T with ||A1|−|B1|| ≤ 1

and A, B is a bipartition of the vertex set of the α-tree Pk v©T then ||A|−|B|| ≤ 1.

Thus, according to Theorem 23, we obtain that Pk v©T admits a super (a, d)-EAT


We denote by T1 the graph Pk v©T . According to Theorem 25, if T is graceful

and k is an even positive integer, then T1 is an α-tree. Thus, by induction it is easy

to see that the graph Tn+1 = Pknvn©Tn, where kn is an even positive integer and vn

is an arbitrary fixed vertex in Tn, is also an α-graph according to Theorem 25.

Moreover, from the construction of the α-tree Tn+1 it follows that |An+1| = |Bn+1|,

where An+1, Bn+1 is the bipartition of the vertex set of Tn+1. Thus, according

to Theorem 26, we get that Tn+1 admits a super (a, d)-EAT labeling for all d ∈

0, 1, 2, 3. We can summarize this results in the following theorem.

Theorem 28. [20] Let k be an even positive integer. Let v be an arbitrary fixed

vertex in tree T and let T1 = Pk v©T . Let n be a positive integer and kn be an even

positive integer. Let vn be an arbitrary fixed vertex in Tn and let Tn+1 = Pknvn©Tn.

The graph Tn+1 is super (a, d)-EAT for all d ∈ 0, 1, 2, 3 if T is a graceful tree.

Notice that for graceful tree T and for different fixed vertices of trees Ti, i =

1, 2, . . . , n, we can find many different (up to isomorphism) α-trees Tn+1 and their

super (a, d)-EAT labelings.


4.4 Certain families of super (a, d)-edge-antimagic

total trees

A caterpillar is a tree with the property that the removal of its endpoints leaves

a path. Rosa [82] proved that all caterpillars admit α-labeling. Figure 1.11 provides

an example of α-labeling of caterpillar on 12 vertices.

A lobster is a tree having a central path H = x0x1 . . . xm−1xm from which ev-

ery vertex has distance at most 2. The removal of the endpoints of lobster leaves

a caterpillar. Figure 4.4 exhibits graceful labelings of two lobsters on 9 vertices.

3 9 1 8

4

7 2

6

5

3 9 1 8

4

7 2 6

5

Figure 4.4: Graceful labelings of lobsters.

Ng in [77] describes graceful labelings of lobsters in which each vertex of the

central path is attached to the centers of exactly two branches and, in addition to

this, each of the vertices x0 and xm is attached to the center of a pendant branch.

Wang, Jin, Lu and Zhang in [97] and Mishra and Panigrahi, in [70], give graceful

labelings to the lobsters having diameter at least five in which the degree of xm is


odd and the degree of the rest of the vertices in H are even.

Chen, Lu and Yeh in [39] give graceful labelings to some classes of lobsters in

which the vertices of the central path are attached to the isomorphic copies of at

most two different branches. Morgan [74] has proved that all lobsters with perfect

matching are graceful.

In [71] and [72], the graceful lobsters have the property that the degree of x0 is

even and the degrees of some (or all) vertices xi, for 1 ≤ i ≤ m, may be odd.

A symmetrical tree ST is a rooted tree in which every level contains vertices of the

same degree. In [28] and [79], it is showed that the symmetrical trees are graceful. In

[86], the graceful symmetrical trees are used for describing Object-oriented software

architecture.

Figure 4.5 depicts the graceful labeling of the symmetrical tree ST . In Figure 4.6

(respectively, Figure 4.7) we exhibit an example of the α-labeling of the tree P2 v©ST

(respectively, P4 v©ST ) obtained using the construction described in Theorem 25,

where ST is the symmetrical tree from Figure 4.5.

4 8 3 7 9 6

10

5 1 2

Figure 4.5: Graceful labeling of a symmetrical tree ST .

The corresponding (3, 2)-EAV labeling of P4 v©ST is shown on Figure 4.8. Ac-

cording to Theorem 3, we can obtain a super (82, 1)-EAT labeling and a super


4 8 3 7 9 6

10

15 11 12

14 18 13 17 19 16

20

5 1 2

Figure 4.6: α-labeling of P2 v©ST .

36 39 37 33 38 34

40

2 1 5

24 28 23 27 29 26

30

15 11 12

6 9 7 3 8 4

10

32 31 35

14 18 13 17 19 16

20

25 21 22

Figure 4.7: α-labeling of P4 v©ST .

(44, 3)-EAT labeling of P4 v©ST .

A banana tree

(a1K1,1, . . . , at−1K1,t−1, atK1,t, at+1K1,t+1, . . . , anK1,n)

denotes the tree obtained by adding a vertex, the apex, to the union of ai copies of the

stars K1,i, and joining the apex to a leaf of each star. Bhat-Nayak and Deshmukh

[29] constructed three new families of graceful banana trees using an algorithmic

labeling proof. They have shown that the following banana trees are graceful:

BK = (K1,1, . . . , K1,t−1, (β + 1)K1,t, K1,t+1, . . . , K1,n),

where 0 ≤ β < t,


10 4 8 16 6 14

2

3 1 9

34 26 36 28 24 30

22

29 21 23

11 17 13 5 15 7

19

18 20 12

27 35 25 33 37 31

39

32 40 38

Figure 4.8: (3, 2)-EAV labeling of P4 v©ST .

BKK = (2K1,1, . . . , 2K1,t−1, (β + 2)K1,t, 2K1,t+1, . . . , 2K1,n),

where 0 ≤ β < t, and

BKKK = (3K1,1, 3K1,2, . . . , 3K1,n).

Murugan and Arumugan [76] additionally showed that any banana tree BKR,

where all stars have the same size, is graceful, by constructing a graceful labeling of

these banana trees.

The regular bamboo trees are rooted trees consisting of the branches, the paths

from the root to the leaves, of equal length, the leaves of which are identified with

leaves of stars of equal size. These were shown to be graceful by Sekar in [85] (see

also [42] and [46]).

The olive trees T(t) are rooted trees with t branches, the ith branch of which

is a path of length i. Abhyankar and Bhat-Nayak [1] gave direct graceful labeling

methods for olive trees.

By F we denote the family of graceful trees that contains caterpillars, symmetrical

trees, lobsters from [39, 70, 71, 72, 74, 77, 97], olive trees, bamboo trees and banana

trees of type BK, BKK, BKKK or BKR.

As the consequences of Theorem 26, 27 and 28 we have the following corollary:

Corollary 7. Let T ∈ F and v be an arbitrary fixed vertex in T .

(i) For an even positive integer k the tree T1 = Pk v©T admits a super (a, d)-EAT


(ii) For every positive integer n the tree Tn+1 = Pknvn©Tn, where kn is a positive

integer and vn is an arbitrary fixed vertex in Tn, admits a super (a, d)-EAT


56

Appendix – Graph-theoretic symbols

E(G) edge set of G

V (G) vertex set of G

|E(G)| = q size of G

|V (G)| = p order of G

(p, q)-graph graph of order p and size q

d(x) degree of vertex x (in G)

N(x) set of all neighbors of the vertex x

δ(G) minimum degree of G

∆(G) maximum degree of G

Pn path on n vertices

Cn cycle on n vertices

Kn complete graph on n vertices

Km,n complete bipartite graph with partite sets

of cardinalities m and n

P (n,m) generalized Petersen graph

T tree

Wn wheel on n + 1 vertices

Fn fan on n + 1 vertices

Fn friendship graph on 2n + 1 vertices

mH union of m disjoint copies of graph H

57

G×H Cartesian product of G and H

Dn prism Cn × P2

An antiprism

G ∪H union of G and H

H ⊆ G graph H is a subgraph of graph G

K1,n star on n + 1 vertices

Pm,n parachute graph

G v©H graph obtained by Koh, Rogers and Tan’s operation

using the graphs G and H

T(t) olive tree with t branches

ST symmetrical tree

BK,BKK, banana trees

BKKK,BKR banana trees

58

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