On k-Edge-magic Halin Graphs Sin-Min Lee, San Jose State University Hsin-hao Su*, Stonehill College Yung-Chin Wang, Tzu-Hui Institute of Technology 41th SICCGTC At Florida Atlantic University March 9, 2010

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.

k-Edge-Magic Graphs A (p,q)-graph G is called k-edge-magic (in short k-EM) if there is an edge labeling l: E(G)  {k,k+1,…,k+q-1}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. If k =1, then G is said to be edge-magic.

Examples: 1-Edge-Magic The following maximal outerplanar graphs with 6 vertices are 1-EM.

Examples: 1-Edge-Magic In general, G may admits more than one labeling to become a k-edge-magic graph with different vertex sums.

Examples: k-Edge-Magic In general, G may admits more than one labeling to become a k-edge-magic graph.

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

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

k-Edge-Magic is periodic Theorem: If a (p,q)-graph G is k-edge-magic then it is pt+k-edge-magic for all t ≥ 0 .

Halin Graphs Definition: Halin graphs are planar connected graphs that consist of a tree and a cycle connecting the end vertices of the tree.

Wheels For n > 3, the wheel on n vertices, Wn is a graph with n vertices x1, x2,..., xn, x1 having degree n-1 and all the other vertices having degree 3.

Wheels – Wn # of vertices: n. # of edges: 2n-2. Necessary condition:

Wheels – Wn‘s Possible k Necessary condition: Let d = gcd(4,n). For t ≥ 1, If n=4t, then d = 4. If n=4t+2, then d = 2. If n=4t+1 or 4t+3, then d = 1. Possible k: If n=4t, then there is no k. If n=4t+2, then k=t+2 or 3t+3 (mod n). If n=4t+1, then k=2t+2 (mod n). If n=4t+3, then k=2t+3 (mod n).

Wheels – not k-EM Theorem: The Halin graph of Wn for n = 4t is not k-edge-magic for all k.

Wheels – W5 Theorem: The Halin graph of W5 is k-edge-magic for all k ≡ 4 (mod 5).

Wheels – W6 Theorem: The Halin graph of W6 is k-edge-magic for all k ≡ 0,3 (mod 6).

Wheels – W7 Theorem: The Halin graph of W7 is k-edge-magic for all k ≡ 5 (mod 7).

Wheels – W9 Theorem: The Halin graph of W9 is k-edge-magic for all k ≡ 6 (mod 9).

Wheels – W2n+1 Theorem: The Halin graph of W2n+1 is k-edge-magic for all k ≡ n+2 (mod 2n+1).

Double Stars Definition: The double star D(m,n) is a tree of diameter three such that there are m appended edges on one ends of P2 and n appended edges on another end.

Double Stars – D(m,n) # of vertices: m+n+2. # of edges: 2(m+n)+1. Necessary condition:

Double Stars – Possible k Necessary condition: Let d = gcd(6,m+n+2). Then, d |12. Possible k: For t ≥ 1, If m+n=6t-4, then k=2 or 3t+1 (mod m+n+2). If m+n=6t-3, then k=2 (mod m+n+2). If m+n=6t-2, then k=2 or 3t+1 (mod m+n+2). If m+n=6t-1, then k=2,t+2,2t+2,3t+2,4t+2,5t+2 (mod m+n+2). If m+n=6t, then k=2 or 3t+3 (mod m+n+2). If m+n=6t+1, then k=2 or 2t+3 or 4t+4 (mod m+n+2).

Double Stars – D(2,2) Theorem: The Halin graph of D(2,2), H(D(2,2)), is k-edge-magic for all k.

Double Stars – D(2,2)

Double Stars – D(2,3) Theorem: The Halin graph of D(2,3) is k-edge-magic for all k ≡ 2 (mod 7).

Double Stars – D(2,4) Theorem: The Halin graph of D(2,4) is k-edge-magic for all k ≡ 2,6 (mod 8).

Double Stars – D(3,3) Theorem: The Halin graph of D(3,3) is k-edge-magic for all k ≡ 2,6 (mod 8).

Spiders – Sp(2,0,2) Theorem: The Halin graph of Sp(2,0,2) is k-edge-magic for all k ≡ 6 (mod 7).

Spiders – Sp(2,0,4) Theorem: The Halin graph of Sp(2,0,4) is k-edge-magic for all k ≡ 7 (mod 9).

Spiders – Sp(3,0,3) Theorem: The Halin graph of Sp(3,0,3) is k-edge-magic for all k ≡ 7 (mod 9).

Spiders – Sp(2,0,5) Theorem: The Halin graph of Sp(2,0,5) is k-edge-magic for all k ≡ 0,5 (mod 10).

Spiders – Sp(3,0,4) Theorem: The Halin graph of Sp(3,0,4) is k-edge-magic for all k ≡ 0,5 (mod 10).

L-Product of Stars with Stars Theorem: The Halin Graph of St(3)xLSt(2) is k-edge-magic for all k ≡ 0 (mod 10).

Spiders – Sp(1n,22) Theorem: The Halin graph of Sp(1n,22) with n = 2i-5 are not k-edge-magic for all k.

Spiders – Sp(11,22) Theorem: The Halin graph of Sp(11,22) is k-edge-magic for all k ≡ 1 (mod 6).

Spiders – Sp(12,22) Theorem: The Halin graph of Sp(12,22) is k-edge-magic for all k ≡ 6 (mod 7).

Spiders – Sp(14,22) Theorem: The Halin graph of Sp(14,22) is k-edge-magic for all k ≡ 7 (mod 9).

Spiders – Sp(15,22) Theorem: The Halin graph of Sp(15,22) is k-edge-magic for all k ≡ 0,5 (mod 10).

Order 8 – Not k-EM Theorem: Among 21 Halin graphs of order 8, the following graphs are not k-edge-magic for all k.

Order 8 – 2-EM and 6-EM Theorem: The following Halin graphs of order 8 is k-edge-magic for all k ≡ 2,6 (mod 8).

Order 8 – 3-EM and 7-EM Theorem: The following Halin graphs of order 8 is k-edge-magic for all k ≡ 3,7 (mod 8).

Order 8 – 3-EM and 7-EM Theorem: The following Halin graphs of order 8 is k-edge-magic for all k ≡ 3,7 (mod 8).

Order 8 – 3-EM and 7-EM Theorem: The following Halin graphs of order 8 is k-edge-magic for all k ≡ 3,7 (mod 8).

Order 8 – 3-EM and 7-EM Theorem: The following Halin graphs of order 8 is k-edge-magic for all k ≡ 3,7 (mod 8).

Order 8 – 3-EM and 7-EM Theorem: The following Halin graphs of order 8 is k-edge-magic for all k ≡ 3,7 (mod 8).

Order 8 – 3-EM and 7-EM Theorem: The following Halin graphs of order 8 is k-edge-magic for all k ≡ 3,7 (mod 8).

Order 8 – 7-EM only Theorem: The following Halin graphs of order 8 is k-edge-magic for k ≡ 7 (mod 8).

Order 8 – 8-EM only Theorem: The following Halin graphs of order 8 is k-edge-magic for k ≡ 7 (mod 8).