On Mod(3)-Edge-magic Graphs

Sin-Min Lee, San Jose State University

Karl Schaffer, De Anza College

Hsin-hao Su*, Stonehill College

Yung-Chin Wang, Tzu-Hui Institute of Technology

6th IWOGL 2010At

University of Minnesota, Duluth

October 22, 2010

Supermagic Graphs

For a (p,q)-graph, in 1966, Stewart[1] defined that a graph labeling is supermagic iff the edges are labeled 1,2,3,…,q so that the vertex sums are a constant.

[1] B.M. Stewart, Magic Graphs, Canadian Journal of Mathematics 18 (1966), 1031-1059.

Magic Square

The classical concept of a magic square of n2 boxes corresponds to the fact that the complete bipartite graph K(n,n) is super magic if n ≥ 3.

Edge-Magic Graphs

Lee, Seah and Tan in 1992 defined that a (p,q)-graph G is called edge-magic (in short EM) if there is an edge labeling l: E(G) {1,2,…,q} such that for each vertex v, the sum of the labels of the edges incident with v are all equal to the same constant modulo p; i.e., l+(v) = c for some fixed c in Zp.

Examples: Edge-Magic

The following maximal outerplanar graphs with 6 vertices are EM.

Examples: Edge-Magic

In general, G may admits more than one labeling to become an edge-magic graph with different vertex sums.

Mod(k)-Edge-Magic Graphs

Let k ≥ 2. A (p,q)-graph G is called Mod(k)-edge-

magic (in short Mod(k)-EM) if there is an edge labeling l: E(G) {1,2,…,q} such that for each vertex v, the sum of the labels of the edges incident with v are all equal to the same constant modulo k; i.e., l+(v) = c for some fixed c in Zk.

Examples

A Mod(k)-EM graph for k = 2,3,4,6, but not a Mod(5)-EM graph.

Examples

The path P4 with 4 vertices is Mod(2)-EM, but not Mod(k)-EM for k = 3,4.

Paths

Theorem: A path P2 is Mod(k)-EM for all k. Proof: There is only one edge. Must be

labeled 1. Theorem: When n > 2, the path Pn is

Mod(k)-EM if and only if k = 2 and n is even.

Notations

For n > 2, let the vertices of Pn be v1, v2, v3, …, vn, where v1 and vn are the end vertices of degree 1, and vi is adjacent to vi+1, for i = 1, 2, …, n-1.

Let the edge joining vertices vi and vi+1 be ei, for i = 1, 2, …, n-1.

Proof

Suppose e1 receives edge label m. Then the vertex v1 is labeled m.

For the vertex v2 to be labeled m as well, edge e2 needs to be labeled 0.

Similarly, the remaining edges need to be labeled by m and 0, alternately.

This is only possible when k = 2 and n is even, in which each vertex labeled 1.

Cubic Graphs

Definition: 3-regular (p,q)-graph is called a cubic graph.

The relationship between p and q is

Since q is an integer, p must be even.

2

3pq

Sufficient Condition

Theorem: If a cubic graph G is Hamiltonian, then it is Mod(3)-EM.

Proof: Note that since G is a cubic graph, p is even. We label all the edges of the cycle by 1, -1

(mod 3) alternatively and the rest edges by 0 (mod 3). It is easy to check that the vertices will be labeled by 0. pkqq mod012

Examples

Cylinder Graphs

Theorem: A cylinder graph CnxP2 is Mod(3)-EM for all n ≥ 3.

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Möbius Ladders

The concept of Möbius ladder was introduced by Guy and Harry in 1967.

It is a cubic circulant graph with an even number n of vertices, formed from an n-cycle by adding edges (called “rungs”) connecting opposite pairs of vertices in the cycle. pkqq mod012

Möbius Ladders A möbius ladder ML(2n)

with the vertices denoted by a1, a2, …, a2n. The edges are then {a1, a2}, {a2, a3}, … {a2n, a1}, {a1, an+1}, {a2, an+2}, … , {an, a2n}. pkqq mod012

Möbius Ladders

Theorem: A Möbius ladder ML(2n) is Mod(3)-EM for all even n ≥ 4.

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Turtle Shell Graphs

Add edges to a cycle C2n with vertices a1, a2, …, an, b1, b2, …, bn such that a1 is adjacent to b1, and ai is adjacent to bn+2-i, for i = 2, …, n. The resulting cubic graph is called the turtle shell graph of order 2n, denoted by TS(2n).

Theorem: The turtle shell graph TS(2n) is Mod(3)-EM for all n ≥ 3.

Turtle Shell Graphs Examples

Coxeter Graphs

For n > 3, we append on each vertex of Cn with a star St(3), and then join all the leaves of the stars by a cycle C2n. We denote the resulting cubic graph by Cox(n).

Note Cox(n) has 4n vertices. Theorem: The Coxeter graph Cox(n) is

Mod(3)-EM for all n ≥ 3.

Coxeter Graph Examples

Corollaries

Corollary: If a cubic graph is Hamiltonian, then it is Mod(3)-EM.

Corollary: Almost all cubic graphs are Mod(3)-EM. pkqq mod012

Issacs Graphs

For n > 3, we denote the graph with vertex set V = { xj, ci,j: i =1,2,3, j = 1, 2, …, n} such that ci,1, ci,2, …, ci,n are three disjoint cycles and xj is adjacent to c1,j, c2,j, c3,j.

We call this graph Issacs graph and denote by IS(n).

Issacs Graphs

Issacs graphs were first considered by Issacs in 1975 and investigated in Seymour in 1979.

They are cubic graphs with perfect matching.

Theorem: The Issacs graph IS(2n) is Mod(3)-EM for an even n ≥ 4.

Issacs Graph’s Inner Cycle

Issacs Graphs Examples

Twisted Cylinder Graphs

Theorem: All twisted cylinder graph TW(n) are Mod(3)-EM.

Remark: Twisted cylinder graph TW(n) is NOT hamiltonian. pkqq mod012

Twisted Cylinder Graphs Ex.

Conjecture

Conjecture[2]: A cubic graph with order p = 4s+2 is Mod(3)-EM.

With the previous examples, this is a reasonable extension of a conjecture by Lee, Pigg, Cox in 1994. pkqq mod012

[2] S-M. Lee, W.M. Pigg, T.J. Cox, On Edge-Magic Cubic Graphs Conjecture, Congressus Numeratium 105 (1994), 214-222.

Sufficient Condition Extended

Theorem: If a cubic graph G of order p has a 2-regular subgraph with p edges, then it is Mod(3)-EM.

Proof: The same labelings work here. pkqq mod012

Mod(2)-EM Classification

(Lee, Su, Wang) Theorem: If a cubic graph G of order p is Mod(2)-EM if and only if it has a 2-regular subgraph with 3p/4 or 3p/4 edges.

Actually, this theorem looks true for all n-regular graphs. The same proof of cubic graphs should apply to n-regular graphs with some minor modifications.

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Degree 3 Vertices

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Necessary Condition

Question: If a cubic graph G of order p is Mod(3)-EM, then it has a 2-regular subgraph with p edges. pkqq mod012

Generalized Petersen Graphs The generalized Petersen graphs P(n,k) were

first studied by Bannai and Coxeter. P(n,k) is the graph with vertices {vi, ui : 0 ≤ i

≤ n-1} and edges {vivi+1, viui, uiui+k}, where subscripts modulo n and k.

(Alspach 1983; Holton and Sheehan 1993) The generalized Petersen graph GP(n,k) is nonhamiltonian iff k = 2 and n ≡ 5 (mod 6).

Generalized Petersen Graphs

Theorem: A generalized Petersen graphs GP(n,k) is Mod(3)-EM for all (n,k) not of the form ( 5 mod(6) , 2 ). pkqq mod012

Petersen Graph Example

Necessary Condition Failed

The Peterson graph shows that the necessary condition is not held since it does not have a path of order 10, but it is a Mod(3)-EM. pkqq mod012

Future Study

Is it possible to find an if and only if condition to classify Mod(3)-EM cubic graphs?

Can we extend the sufficient condition to n-regular graphs? pkqq mod012