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