1. 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

2. 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.

3. Magic Square The classical concept of a magic square of n 2 boxes corresponds to the fact that the complete bipartite graph K(n,n) is super magic if n ≥ 3.

4. 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 Z p.

5. Examples: Edge-Magic The following maximal outerplanar graphs with 6 vertices are EM.

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

7. 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 Z k.

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

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

10. Paths Theorem: A path P 2 is Mod(k)-EM for all k. Proof: There is only one edge. Must be labeled 1. Theorem: When n > 2, the path P n is Mod(k)-EM if and only if k = 2 and n is even.

11. Notations For n > 2, let the vertices of P n be v 1, v 2, v 3, …, v n, where v 1 and v n are the end vertices of degree 1, and v i is adjacent to v i+1, for i = 1, 2, …, n-1. Let the edge joining vertices v i and v i+1 be e i, for i = 1, 2, …, n-1.

12. Proof Suppose e 1 receives edge label m. Then the vertex v 1 is labeled m. For the vertex v 2 to be labeled m as well, edge e 2 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.

13. 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.

14. 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.

15. Examples

16. Cylinder Graphs Theorem: A cylinder graph C n xP 2 is Mod(3)-EM for all n ≥ 3.

17. 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.cubiccirculant grapheven number cycle

18. Möbius Ladders A möbius ladder ML(2n) with the vertices denoted by a 1, a 2, …, a 2n. The edges are then {a 1, a 2 }, {a 2, a 3 }, … {a 2n, a 1 }, {a 1, a n+1 }, {a 2, a n+2 }, …, {a n, a 2n }.

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

20. Turtle Shell Graphs Add edges to a cycle C 2n with vertices a 1, a 2, …, a n, b 1, b 2, …, b n such that a 1 is adjacent to b 1, and a i is adjacent to b n+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.

21. Turtle Shell Graphs Examples

22. Coxeter Graphs For n > 3, we append on each vertex of C n with a star St(3), and then join all the leaves of the stars by a cycle C 2n. 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.

23. Coxeter Graph Examples

24. Corollaries Corollary: If a cubic graph is Hamiltonian, then it is Mod(3)-EM. Corollary: Almost all cubic graphs are Mod(3)-EM.

25. Issacs Graphs For n > 3, we denote the graph with vertex set V = { x j, c i,j : i =1,2,3, j = 1, 2, …, n} such that c i,1, c i,2, …, c i,n are three disjoint cycles and x j is adjacent to c 1,j, c 2,j, c 3,j. We call this graph Issacs graph and denote by IS(n).

26. 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.

27. Issacs Graph’s Inner Cycle

28. Issacs Graphs Examples

29. Twisted Cylinder Graphs Theorem: All twisted cylinder graph TW(n) are Mod(3)-EM. Remark: Twisted cylinder graph TW(n) is NOT hamiltonian.

30. Twisted Cylinder Graphs Ex.

31. 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.

32. 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.

33. 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.

34. Degree 3 Vertices

35. 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.

36. 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 {v i, u i : 0 ≤ i ≤ n-1} and edges {v i v i+1, v i u i, u i u i+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).

37. 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 ).

38. Petersen Graph Example

39. 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.

40. 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?