Edge-magic Indices of Stars Sin-Min Lee, San Jose State University Yong-Song Ho and Sie-Keng Tan, Nat’l Univ. of Singapore Hsin-hao Su *, Stonehill College 43rd Southeastern International Conference on Combinatorics, Graph Theory, and Computing at Florida Atlantic University March 8, 2012

Supermagic Graphs For a (p,q)-graph, in 1966, Stewart defined that a graph labeling is supermagic iff the edges are labeled 1,2,3,…,q so that the vertex sums are a constant.

Edge-Magic Graphs A (p,q)-graph G is called edge-magic (in short EM) if there is a bijective 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.

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.

Edge-Splitting Extension Graphs For a (p,q)-graph G=(V,E), we can construct a graph SPE(G,f), namely, edge-splitting extension graph as follow: for each edge e in E, we associate a set of f(e) parallel edges. If f is a constant map, f(e) = k for some integer k in N, the we denote SPE(G,f) as G[k]. (Note: G[1]=G.)

Edge-Magic Index It is easy to see that for any (p,g)-graph G, G[2p] is edge-magic. The set {k | G[k] is edge-magic} is denoted by IM(G). The smallest number in IM(G) is called the edge-magic index of G, denoted by emi(G).

Necessary Condition A necessary condition for a (p,q)- multigraph G to be edge-magic is Proof: The sum of all edges is Every edge is counted twice in the vertex sums.

Upper Bounds of emi

Basic Number Theory Proposition: Let d = gcd(a,m). ax = b has a solution in Z m iff d | b. Moreover, if d | b, then there are exactly d solutions in Z m.

Star Graphs Definition: A star graph, St(n), is a graph with n+1 vertices where one vertex, called center, is of degree n and others, called leaves, are of degree 1. It is obvious that St(n) is not edge- magic. Thus the edge-magic index is greater or equal to 2.

Upper Bound Theorem: The upper bound of the edge- magic index of St(n) is n+1. Proof: If n is odd, then all the vertices are of odd degree, then the edge-magic index ≤ n+1. If n is even, then St(n) has odd number of vertices, then the edge-magic index ≤ n+1.

Vertex Labels

Necessary Condition for k

Edge-Magic Index of Stars

If k is even

If k is even (continued)

An example when k is even Consider St(4). By solving the necessary condition, the smallest k is 6. Pair the numbers 1,2,…,24 into 12 pairs as (1,24), (2,23), (3,22), …, (11,14), (12,13). Label the 6 edges joining u and v 1 by 1, 24, 2, 23, 3, 22. The sum of v 1 is 45. Label the 6 edges joining u and v 2 by 4, 21, 5, 20, 6, 19. The sum of v 2 is 45.

St(4) Label the 6 edges joining u and v 3 by 7, 18, 8, 17, 9, 16. The sum of v 3 is 45. Label the 6 edges joining u and v 4 by 10, 15, 11, 14, 12, 13. The sum of v 4 is 45. The sum of u is 180(=45×4).

If k is odd

If k is odd (continued)

An example when k is odd Consider St(9). By solving the necessary condition, the smallest k is 5. Group the numbers 1,2,…,27 into 9 triples as (1,18,23), (2,13,27), (3,17,22), (4,12,26), (5,16,21), (6,11,25),(7,15,20), (8,10,24), (9,14,19).

An example when k is odd Pair the remaining 18 numbers 28,29,…,45 into 9 pairs as (28,45), (29,44), (30,43), (31,42), (32,41), (33,40),(34,39), (35,38), (36,37). Label the 5 edges joining u and v 1 by 1, 18, 23, 28, 45. The sum of v 1 is 115. Label the 5 edges joining u and v 2 by 2, 13, 27, 29, 44. The sum of v 2 is 115.

St(9) Label the 5 edges joining u and v 3 by 3, 17, 22, 30, 43. The sum of v 3 is 115. Label the 5 edges joining u and v 4 by 4, 12, 26, 31, 42. The sum of v 4 is 115. Label the 5 edges joining u and v 5 by 5, 16, 21, 32, 41. The sum of v 5 is 115. Label the 5 edges joining u and v 6 by 6, 11, 25, 33, 40. The sum of v 6 is 115.

St(9)

If n is odd