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

atFlorida 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 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.

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.

pqq mod01

2

11 qq

Upper Bounds of emi

Theorem: For a (p,g)-graph G, if p is not even or q is not odd, then the edge-magic index of G is less than or equal to p.

Proof: Label {k, k+q, k+2q, …, k+(p-1)q} on a set of

parallel edges of G[p]. The sum of these p edges is .

Upper Bounds of emi

Theorem: For a (p,g)-graph G, if all vertices are odd (or even) degrees, then the edge-magic index of G is less than or equal to p.

Proof: Label {1+pk, 2+pk, …, p+pk} on a set of parallel

edges of G[p]. The sum of these p edges is . The sum of a vertex is .

Upper Bounds of emi

Theorem: For a (p,g)-graph G, if p divides q, then the edge-magic index of G is less than or equal to 2.

Proof: If p is odd, label by pairs (1,q-1), (1,q-1), (2,q-2),

(2,q-2), …, (, ), (, ), (q,q). If p is even, label by pairs (1,q-1), (1,q-1), (2,q-

2), (2,q-2), …, (, ), (, ), (q,q).

Upper Bounds of emi

Theorem: The edge-magic index of a regular graph G is less than or equal to 2.

Proof: Let the degree of a vertex is r. Label by pairs

(1,pr), (2,pr-1), (3,pr-2), …, (, ). Then, the sum of a vertex is r(1+pr).

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

For St(n)[k], since the center, u, is adjacent to every edge, the sum is .

For each leaf, vi, if the labeling is edge-magic, their labels are all the same and summed up to . Because there are n leaves, the label is .

Necessary Condition for k

If the labeling is edge-magic, then must be congruence to modulo n+1.

Thus, k must satisfy

Edge-Magic Index of Stars

Theorem: The edge-magic index of St(n) is the smallest positive integer k satisfied

Proof: We provide two magic labelings for even or

odd k satisfied the above condition.

If k is even

Pair the numbers 1,2,…,kn into pairs as (1,kn), (2,kn-1), (3,kn-2), …, (, ), (, ).

Note that the sum of each pair is always kn+1.

If k is even (continued)

Use the first pairs to label k edges between u and v1.

Then use the next pairs to label k edges between u and v2.

Continue the process until label every edge.

If k is even (continued)

The sum of vi for all i is . The sum of u is . Since k satisfies the necessary condition,

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 v1 by 1, 24, 2, 23, 3, 22. The sum of v1 is 45.

Label the 6 edges joining u and v2 by 4, 21, 5, 20, 6, 19. The sum of v2 is 45.

St(4)

Label the 6 edges joining u and v3 by 7, 18, 8, 17, 9, 16. The sum of v3 is 45.

Label the 6 edges joining u and v4 by 10, 15, 11, 14, 12, 13. The sum of v4 is 45.

The sum of u is 180(=45×4).

If k is odd

Group the numbers 1,2,…,3n into n group of 3 numbers as {} for t = 0,1,2,…, and {} for t = 1,2,…, .

Note that the sum of each group is .

Note that the sum of each pair is always kn+1.

If k is odd (continued)

Pair the numbers 3n+1,3n+2,…,kn into pairs as (3n+1,kn), (3n+2,kn-1), (3n+3,kn-2), (3n+4,kn-3), ….

Note that the sum of each pair is (3n+1)+kn.

If k is odd (continued)

Use a group of 3 numbers with pairs to label k edges between u and v1.

Then use another group of 3 numbers with the next pairs to label k edges between u and v2.

Continue the process until label every edge.

If k is odd (continued)

The sum of vi for all i is . The sum of u is . Since k satisfies the necessary condition,

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 v1 by 1, 18, 23, 28, 45. The sum of v1 is 115.

Label the 5 edges joining u and v2 by 2, 13, 27, 29, 44. The sum of v2 is 115.

St(9)

Label the 5 edges joining u and v3 by 3, 17, 22, 30, 43. The sum of v3 is 115.

Label the 5 edges joining u and v4 by 4, 12, 26, 31, 42. The sum of v4 is 115.

Label the 5 edges joining u and v5 by 5, 16, 21, 32, 41. The sum of v5 is 115.

Label the 5 edges joining u and v6 by 6, 11, 25, 33, 40. The sum of v6 is 115.

St(9)

Label the 5 edges joining u and v7 by 7, 15, 20, 34, 39. The sum of v7 is 115.

Label the 5 edges joining u and v8 by 8, 10, 24, 35, 38. The sum of v8 is 115.

Label the 5 edges joining u and v9 by 9, 14, 19, 36,37. The sum of v9 is 115.

The sum of u is 1035(=115×4).

If n is odd

Corollary: The edge-magic index of St(n) is the smallest positive integer k satisfied

Proof: .