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Super edge-graceful labelings for total stars and total cycles

Super edge-graceful labelings for total stars and total cycles. Abdollah Khodkar Department of Mathematics University of West Georgia www.westga.edu/~akhodkar Joint work with: Kurt Vinhage, Florida State University. Overview. Edge-graceful labeling. 2. Super edge-graceful labeling.

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Super edge-graceful labelings for total stars and total cycles

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

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

  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. 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. An Edge-graceful labeling for K4 minus an edge 1 4 0 1 3 5 2 2 3

  6. Theorem: (Lo 1985) A necessary condition for a graph of order p and size q to be edge-graceful is that p divides (q2+q-(p(p-1)/2)). That is, q(q +1) ≡ p(p-1)/2 (mod p). 6

  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 q2+q-(p(p-1)/2)=p2+p-(p(p-1)/2). Therefore, p(p-1)/2=kp for some positive integer k. This implies p=2k+1. 7

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

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

  10. Theorem: Lee, Lee and Murty (1988) If G is a graph of order p≡ 2 (mod 4), then G is not edge-graceful. 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.

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

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

  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/2if q is even and with 0, ±1, ±2,…,±(q-1)/2if q is odd. If the vertex sums are ±1, ±2,…,±p/2when p is even and 0, ±1, ±2,…,±(p-1)/2when p is odd, then G is super edge-graceful.

  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. Theorem: Super edge-graceful trees of odd order are edge-graceful. S.-M. Lee and Y.-S. Ho (2007): All trees of odd order with three even vertices are super edge-graceful. 14

  15. S. Cichacz, D. Froncek, W. Xu and A. Khodkar (2008): All paths Pnexcept P2and P4and all cycles except C4and C6are super edge-graceful. A. Khodkar, R. Rasi and S.M. Sheikholeslami (2008): The complete graph Kn issuper edge-gracefulfor all n ≥ 3, n ≠ 4.

  16. A. Khodkar, S. Nolen and J. Perconti (2009): All complete bipartite graphs Km,n are super edge-graceful except for K2,2, K2,3, and K1,n if n is odd. A. Khodkar (2009): All complete tripartite graphs are super edge-graceful except for K1,1,2.

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

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

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

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

  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 21

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

  23. Total cycle T(C8)

  24. Edge Labels: ±1, ±2, ± 3, … , ± 12 Vertex Labels: ±1, ±2, ± 3, …, ± 8 4 -4 -12 12 5 3 -5 -8 8 -3 11 -11 -6 6 -1 2 -2 1 10 -10 -7 7 9 -9 SEGL of total cycle T(C8)

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

  26. SEGL for T(St(16))

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

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

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

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

  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. An Open Problem: Super edge-gracefulness of disjoint union of four cycles. Edge Labels=Vertex Labels={1, -1, 2, -2} 0 -1 -1 1 2 1 3 1 Hence, C4 is not super edge-graceful. 33

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

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

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

  36. 37

  37. Thank You 38

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