1. Abdollah Khodkar Department of Mathematics University of West Georgia www.westga.edu/~akhodkar Joint work with: Kurt Vinhage, Florida State University Super edge-graceful labelings for total stars and total cycles

2. 2 Overview 1.Edge-graceful labeling 3. Super edge-graceful labeling of total stars 2. Super edge-graceful labeling 4. Super edge-graceful labeling of total cycles 5. An open problem

3. 3 Edge-graceful labeling S.P. Lo (1985) introduced edge-graceful labeling. A graph G of order p and size q is edge-graceful if the edges can be labeled by 1, 2, …, q such that the vertex sums are distinct (mod p).

4. 4 Edge-graceful labeling p=4 So vertex labels are 0, 1, 2, 3 q=5 So edge labels are 1, 2, 3, 4, 5 4 1 3 5 2

5. 5 An Edge-graceful labeling for K 4 minus an edge 1 4 0 1 3 5 2 2 3

6. 6 Theorem: (Lo 1985) A necessary condition for a graph of order p and size q to be edge-graceful is that p divides (q 2 +q-(p(p-1)/2)). That is, q(q +1) ≡ p(p-1)/2 (mod p).

7. 7 Corollary: No cycle of even order is edge-graceful. Proof: In a cycle of order p we have q=p. By the Theorem, p divides q 2 +q-(p(p-1)/2)=p 2 +p-(p(p-1)/2). Therefore, p(p-1)/2=kp for some positive integer k. This implies p=2k+1.

8. 8 Proof: Let p=2k, then q=2k-1. So (2k-1)(2k)-2k(2k-1)/2=2km. Hence, 2k-1=2m, a contradiction. Corollary: There is no edge-graceful tree of even order.

9. 9 Corollary: A complete graphs on p vertices is not edge- graceful, if p ≡ 2 (mod 4). Corollary: Petersen graph is not edge-graceful. Corollary: A complete bipartite graph K m,m is not edge-graceful.

10. 10 Conjecture: Kuan, Lee, Mitchem and Wang (1988) Every odd order unicyclic graph is edge-graceful. Conjecture: Sin-Min Lee (1989) Every tree of odd order is edge-graceful. Theorem: Lee, Lee and Murty (1988) If G is a graph of order p ≡ 2 (mod 4), then G is not edge-graceful.

11. A New Labeling 4 1 2 -4 -3 3 -2

12. A New Labeling 4 1 2 -4 -3 3 -2 2 -2 3 -3 1

13. 13 Super edge-graceful labeling J. Mitchem and A. Simoson (1994): Consider a graph G with p vertices and q edges. We label the edges with ±1, ±2,…,±q/2 if q is even and with 0, ±1, ±2,…,±(q-1)/2 if q is odd. If the vertex sums are ±1, ±2,…,±p/2 when p is even and 0, ±1, ±2,…,±(p-1)/2 when p is odd, then G is super edge-graceful.

14. 14 J. Mitchem and A. Simoson (1994): If G is super edge-graceful and p | q, if q is odd, or p | q+1, if q is even, then G is edge-graceful. S.-M. Lee and Y.-S. Ho (2007): All trees of odd order with three even vertices are super edge-graceful. Theorem: Super edge-graceful trees of odd order are edge-graceful.

15. 15 S. Cichacz, D. Froncek, W. Xu and A. Khodkar (2008): All paths P n except P 2 and P 4 and all cycles except C 4 and C 6 are super edge-graceful. A. Khodkar, R. Rasi and S.M. Sheikholeslami (2008): The complete graph K n is super edge-graceful for all n ≥ 3, n ≠ 4.

16. 16 A. Khodkar, S. Nolen and J. Perconti (2009): All complete bipartite graphs K m,n are super edge-graceful except for K 2,2, K 2,3, and K 1,n if n is odd. A. Khodkar (2009): All complete tripartite graphs are super edge-graceful except for K 1,1,2.

17. 17 A. Khodkar and Kurt Vinhage (2011): Total stars and total cycles are super edge-graceful. Lee, Seah and Tong (2011): Total cycles (T(C n )) are edge-graceful if and only if n is even.

18. 18 Stars Star with 5 vertices: St(5)

19. 19 Total Stars T(St(5))

20. 20 Total Stars T(St(5)) 5 6 -3 -2 1 4 Edge Labels: ±1, ±2, ± 3, ± 4, ± 5, ± 6 Vertex Labels: 0, ±1, ±2, ± 3, ± 4 -5-6 2 3 -4

21. 21 SEGL for T(St(2n+1)) SEGL for T(St(9)) Edge Labels: ±1, ±2, ± 3, …, ± 12 Vertex Labels: 0, ±1, ±2, ± 3, …, ± 8

22. 22 SEGL for T(St(10)) SEGL for T(St(2n)) Edge Labels: 0, ±1, ±2, ± 3, …, ± 13 Vertex Labels: 0, ±1, ±2, ± 3, …, ± 9

23. 23 Total cycle T(C 8 )

24. 24 SEGL of total cycle T(C 8 ) Edge Labels: ±1, ±2, ± 3, …, ± 12 Vertex Labels: ±1, ±2, ± 3, …, ± 8 -8 2 8 1 -2 -3 4 3 -12 5 6 -11 7 -10 9 -9 10 -4 12 11 -5 -6 -7

25. 25 SEGL of total cycle T(C n ) SEGL for T(St(16)) Edge Labels: ±1, ±2, ± 3, …, ± 24 Vertex Labels: ±1, ±2, ± 3, …, ± 16

26. 26 SEGL for T(St(16))

27. 27 SEGL of total cycle T(C n )), n ≡ 0 (mod 8)

28. 28 SEGL for the Union of Vertex Disjoint of 3-Cycles -3 3 0 2 -2 1 -4 4 0 3 -2 4 3 -4 1 2 Edge labels and vertex labels are 0, ±1, ±2, ±3, ±4

29. 29 SEGL for the Union of Vertex Disjoint of 3-Cycles Edge labels and vertex labels are ±1, ±2, ±3, ±4, ±5, ±6 6 -5 5 -6 -4 1 3 -2 -3 2 4 1 -6 -4 5 2 -2 -3 -5 3 6 4

30. 30 c -b -a -c a b Let a + b + c = 0.

31. 31 A. Khodkar (2013): The union of vertex disjoint 3-cycles is super edge-graceful. Example: The union of fifteen vertex disjoint 3-cycles is Super edge graceful.

32. 32

33. 33 An Open Problem: Super edge-gracefulness of disjoint union of four cycles. 1 1 2 0 1 3 Edge Labels=Vertex Labels={1, -1, 2, -2} Hence, C 4 is not super edge-graceful.

34. 34 -4 4 1 2 -3 3 -2 Edge Labels=Vertex Labels={1, -1, 2, -2, 3, -3, 4, -4} 2 -2 1 3 4 -3 -4 Hence, the disjoint union of two 4-cycles is SEG. Disjoint union of two 4-cycles

35. 35 Is the disjoint union of three 4-cycles SEG? Edge Labels=Vertex Labels={±1, ±2, ±3, ±4, ±5, ±6}

36. 36 An Open Problem: The disjoint union of m 4-cycles is super edge-graceful if m>3.

37. 37

38. 38 Thank You