1. 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

2. 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.

3. 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.

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

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

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

7. 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.

8. 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.

9. 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 .

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

11. 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.

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

13. 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).

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

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

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

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

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

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

20. 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.

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

22. 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 (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).

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

24. Double Stars – D(2,2)

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

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

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

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

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

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

31. 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).

32. 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).

33. 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).

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

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

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

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

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

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

40. 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).

41. 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).

42. 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).

43. 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).

44. 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).

45. 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).

46. 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).

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

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