Koc_Arkut

10.12.2003 Koc University 1

Conjectures and Challengesin

Graph Labelings

Ibrahim Cahit ArkutGirne American University


Presentation Plan

• Cordial graphs• Graceful trees• Harmonious graphs• Super magic trees (Beta-

magic)


Challenges and Conjectures

All trees (arithmetic progression)(Lee, 2003)

Friendly index set of trees

Cordial

All trees

(Cahit, 2000)

Trees with diameter 4Beta-Magic

All trees

(Graham, Sloane, 1980)

Trees with diameter 4Harmonious

All trees

(Ringel-Kotzig,1963)

Trees with diameter 6 and lobsters

Graceful


Tree Packing Problem(Problem 25)

Gerhard Ringel(1963)


Origin of the graph labeling problems

• Problem 25. (Ringel, 1963) If T is a tree on m edges, then T6K2m+1.

2 3

1

5 2

4 3

1

=K5= 5 x

(Kotzig, 1965)Cyclically decomposed !

Thm. (Rosa). If T is a graceful tree on m edges, then T6K2m+1.


Definition

• Label the vertices/edges of a graph G with integers from a set S with the property P’ so that when induce edge/vertex labels computed by the rule R imposed another property P on the edge/vertex set.

e.g., (a) S={1,2,…,n}, {0,1}, {2,4,…,2k}, {a,b,c,…} or {-1,0,+1} etc.,(b) P or P’ may be distinct integers, equitably used numbers, etc.,(c) R addition, modular addition or absolute difference of adjacent

vertex labels, edge sum at a vertex etc.,


Graceful, Harmonious and Cordial

• f: V T N={0,1,…,e} and f({u,v})=6f(u)-f(v)6, if induce edge labels N={1,2,…,e} then f is called a graceful labeling.

• f: V TN=Ze and f({u,v})=f(u)+f(v), if induce edge labels N=Ze then f is called a harmonious labeling.

• f: V TN={0,1} and if N={0,1} with 6vf(1)-vf(0) 6b1 and 6ef(1)-ef(0) 6b1 then f is called a cordial labeling.


Reason to non-existence results(according to A. Rosa)

• G has “too many vertices” and “not enough edges”

• G has “too many edges”• G has “ the wrong parity”*

Thm. If every vertex has even degree and 6E(G)6ª1,2 (mod 4) then G is not graceful.


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SOME GRACEFUL

SOME UNGRACEFUL GRAPHS


Computational results

• All trees at most 27 vertices are graceful.(Aldred and McKay).

• Decide whether a graph admits a cordial labeling is NP-complete.(Cairne and Edwards).


Probabilistic results

• Almost all random graphs are cordial.(Godbole, Miller and Ramras).

• Almost all graphs are not graceful.(Erdös).

• Almost all graphs are not harmonious.(Graham and Sloane).


Cordial Labeling

Ibrahim Cahit (1986)


Trees are cordial

• Proof 1. Mathematical induction on n.

• Proof 2. Horce-race labeling algorithm

0

1

1 0

1

0

1

0

1

0

1

322110e(1)

221100e(0)

322210v(1)

332111v(0)


k-equitable labeling of graphsCahit (1990)

• For any positive integer k, assign vertex labels from {0, 1, . . . , k – 1} s.t.

• (1) the number of vertices labeled with i and the number of vertices labeled with j differ by at most one and

• (2) the number of edges labeled with i and the number of edges labeled with j differ by at most one.

• G(V,E) is graceful if and only if it is |E| + 1-equitable and is cordial if and only if it is 2-equitable.


Some results• An Eulerian graph with qTk (mod 2k)

edges is not 3-equitable.• (Szanizló) Cn is k-equitable iff k satisfies:

n∫k; if kª2,3 (mod 4) then n∫k-1if kª2,3 (mod 4) then nTk (mod 2k).

• (Speyer,Szanizló) All trees are 3 equitable.Conjecture (Cahit, 1990): All trees are k-

equitable.


Randomly Cordial Graphs(Chartrand, Min Lee, Zhang,2002)

• Thm. Connected graph G of order nr2 is randomly cordial iff n=3 and G=K3or n is even and G=K1,n-1

Proof: Very lengthy (17 pages)

n=3andK3

K1,n-1,n=even


Friendly Index Set of Cordial Graphs

• f: V(G)TA induces F*:E(G) TA defined f*(xy)=f(x)+f(y), for each xyeE(G).

• For ieA, vf(i)=card{veV(G):f(v)=i} and ef(i)=card{eeE(G):f*(e)=i}.

• Let c(f)={6ef(i)-ef(j)6:(i,j)eAxA}• f of G is said to be A-friendly if 6vf(i)-vf(j)6b1 for all

(i,j)eAxA• If c(f) is a (0,1)-matrix for an A-friendly labeling f

then f is said to be A-cordial.• A=Z2 T FI(G)={6ef(0)-ef(1)6:f is Z2-friendly}


Friendly index sets of trees

• FI(K1,n)={1} if n is odd and {0,2} otherwise.• Complete binary tree with depth 1 has

FI={0,2} and FI={0,2,…,2d+1-4} for depthr2.

• Conjecture (Lee and Ng, 2004). The numbers in FI(T) for any tree T forms an arithmetic progression.


Examples

• Thm. For any G with q edges

FI(G)m{0,2,…,q} if q is even

FI(G)m{1,3,…,q} if q is odd

0 0

1 1

0

1

0

1 1 0

0 1

11 1

1

0

011

1

00 1

1 1 0 1 0 1

FI(K1,3)={1}

FI(K2,2)={0,4}


Example (complete binary tree){0,1,1,0,0,1}

{0,0,0,1,0,0} {1,1,0,1,1,1}

{0,0,0,0,1,1} {1,1,1,1,1,1} {0,0,0,0,0,0} {1,1,1,1,0,0}

{0,1,1,1,0,1} {1,0,1,1,1,0}

{0,0,0,1,1,1} {1,1,1,0,1,1} {1,1,0,1,1,1} {0,0,1,0,1,1}

e(1)=3,e(0)=3 and e(1)-e(0)=0 FI(T2)={0,2,4}


Graceful Labeling

AlexandreRosa(1965)


Alpha-labeling of graphs (Rosa)

• If f is an graceful labeling and there exists mœ{1,2,…,n}s.t. for an arbitrary edge (x,y) of G either f(x)bm and f(y)>m or f(x)>m and f(y)bmholds.


Strong graceful trees

• If f is an graceful labeling and for every x,y,zœV(T) with (x,y),(y,z)œE(T)either f(x)<f(y) and f(y)>f(z) or f(x)>f(y)and f(y)<f(z) holds then f is called ordered.

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Conjecture (Cahit, 1980): All trees are ordered graceful.


Graceful trees

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All caterpillar trees are graceful

Conjecture (Bermond 1979): All lobster trees are graceful.


Rotatability

• A graceful tree is called rotatable if it is possible to assign smallest label to an arbitrary vertex.

• Paths and a class of caterpillars are rotatable. Unrotatable trees

Conjecture:

There exists no trees with more than two unrotatable vertices.


Small rotatable trees

• T(p,1) is rotatable for pT3,11 (mod 12) and pT0 (mod 2).

• T(p,r) is rotatable for p,r=even.

• T(p,r) is not rotatable for p=odd and r=even.

• T(p,r) is rotatable for p,r=odd and (p+r)/2=odd.

p vertices

rvertices

T(p,r)


Spiral canonic labeling

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29

38

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1 n 2 n-1 3 n-2

(1) All trees with diameter four are graceful (Cahit, 1985)

(2) All trees with diameter five are graceful (Hrnciar, Haviar, 2001)


An application of spiral-labeling

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813

3631 9 14

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434401153963338

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15291628172718261925202421

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START

FINISH

Graceful lobster!


Trees with diameter r 6Rooted trees where the roots have odd degree and the lengths of thepaths from the root to the leaves differ by at most one and all the internal vertices have the same parity.

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17181920

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383940

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3 24 44 23 4 25 42 21

43 22 626 42 215 27 41

7 2841 20

36 15 12 33

n=46 , m=2 (mod 4)

Classify them as:

(a) TR(o),I(e) ,

(b) TR(o),I(o) ,

(c) TR(e),I(e) ,

(d) TR(e),I(o)


Harmonious Labeling

Ron L. Graham

and

Neil J. A. Sloane(1980)


Harmonious trees

• f is harmonious labeling of G with e edges if it is possible to label vertices with distinct elements f(x) of Ze, s.t. f(x,y)=f(x)+f(y).

• If G is a tree exactly one vertex label repeated.

Conjecture: All trees are harmonious.


Harmonious labeling of p-stars

• A p-star is a star tree in which each edge is a path of length k.

Challenge: Harmonious labeling of p-stars, where p=even and k>2.

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11

6

75

4

31

20 181614

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191715

mod 20

Harmonious labeling of T(2,10)


Magic Trees and Graphs

Anton Kotzig and Alex Rosa, 1970


Magic (edge) labeling

• f: V(G)(E(G)T{1,2,…,6V(G)6(6E(G)6} is edge-magic if f(x)+f(y)+f(xy)=C for any xyeE(G).

• f is called super-magic if additionally f(V(G))={1,2,…, 6V(G)6} and f(E(G))={6V(G)6+1,…, 6V(G)6(6E(G)6}

• Paths and caterpillars are edge-magic. (Kotzig and Rosa, 1970).

• Complete binary trees are edge-magic. (Cahit, 1980).


Alpha-magic (or consecutive magic)

• A(T,v): set of vertices even distance from v.

• B(T,v): set of vertices odd distance from v.

• If f(A(T,v))={1,2,…,6A(T,v)6}

andf(B(T,v))={6A(T,v)6+1,…,6V(T)6}

then T is called alpha-magic.

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A(T,1)={1,2,3,...,10}

B(T,1)={11,12,....,31}

Complete binary trees are alpha magic


Beta-magic trees (Cahit 2003)

• A tree T is beta-magic if it is super-magic and its vertices partitioned into three color classes C1,C2,C3 of consecutive labels.

• All alpha magic trees are beta magic (reverse is not true).

• All comets (2-stars) are beta magic.

Conjecture: All trees are beta-magic.


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C=19

No alpha-magic labeling exits!


Geometric representation

Vertex labels 1,2,....,k

Vertex labelsk+p+1,k+p+2,...,k+p+r

Vertex labelsk+1,k+2,...,k+p

L1

L2

consecutive edgelabels


Beta-magic diameter four trees

1 2 4 5 6 7 2m 2m+1 2m+2 2m+r

2m+r+1 2m+r+2 2m+r+3 n=2m+r+4

T(m,m,r)

3

2m+r+3

4m+2r+5

4m+2r+4

13 14 15 16

321 121110987654

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1617 18 19 20 21

22 2324

25 26 27

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29

Beta-magic labeling of T(4,4,4) tree

Challenge: T(q1,q2,,,,,qp) is an open-problem for pr4.


Beta-magic labeling of T(2,2,2,2,2,2,2,2), qi=2, 1bib8

1812

9 1 4 5 3 7 13 14

20

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21 22 23 24 25

11

pr9 ???


Mile-stones• A. Rosa, On certain valuations of the vertices of a

graph, in: Theory of Graphs (Proc. Internat. Sympos. Rome 1965), Gordon and Breach, N.Y.-Paris 1967, pp.349-355.

• A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canadian Math. Bull.,13(4),(1970),451-461.

• R. L. Graham, N.J.A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Disc. Math.,1(1980), pp.382-404.

• I. Cahit, Cordial graphs: A weaker version of graceful and harmonious graphs, Ars Combinatoria, 23(1987),pp.201-208.

• J. Gallian, A dynamic survey of graph labeling, October 2003, (http://www.combinatorics.org), DS6.

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