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On Mod(3)-Edge -magic Graphs

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

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  1. 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

  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 n2 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 Zp.

  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 Zk.

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

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

  10. 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.

  11. 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.

  12. 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.

  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 CnxP2 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 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.

  18. 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}.

  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 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.

  21. Turtle Shell Graphs Examples

  22. 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.

  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 = { 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).

  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 {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).

  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?

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