On Mod(3)-Edge -magic Graphs

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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 2010 At University of Minnesota, Duluth October 22, 2010. Supermagic Graphs.

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On Mod(3)-Edge-magic Graphs

Sin-Min Lee, San Jose State University

Karl Schaffer, De Anza College

Hsin-hao Su*, StonehillCollege

Yung-Chin Wang, Tzu-Hui Institute of Technology

6th IWOGL 2010

At

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.
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.

Cylinder Graphs

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

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

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

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}.

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

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.
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.
Corollaries

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

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

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,nare three disjoint cycles and xjis 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.
Twisted Cylinder Graphs

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

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

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.

[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.

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.

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.

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 ).

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.

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?