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 1. 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 2 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). 3 Edge-graceful labeling p=4 q=5 So vertex labels are 0, 1, 2, 3 So edge labels are 1, 2, 3, 4, 5 4 1 3 5 2 4 An Edge-graceful labeling for K4 minus an edge 1 1 2 4 3 0 5 2 3 5 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 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 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. 8 Corollary: A complete graphs on p vertices is not edgegraceful, if p ≡ 2 (mod 4). Corollary: A complete bipartite graph Km,m is not edge-graceful. Corollary: Petersen graph is not edge-graceful. 9 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. 10 A New Labeling 4 -2 2 -1 1 -3 -4 3 A New Labeling 2 4 1 2 3 1 -2 -1 -3 -2 -4 3 -1 -3 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. 13 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 S. Cichacz, D. Froncek, W. Xu and A. Khodkar (2008): All paths Pn except P2 and P4 and all cycles except C4 and C6 are super edge-graceful. A. Khodkar, R. Rasi and S.M. Sheikholeslami (2008): The complete graph Kn is super edge-graceful for all n ≥ 3, n ≠ 4. 15 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. 16 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. 17 Stars Star with 5 vertices: St(5) 18 Total Stars T(St(5)) 19 Total Stars -2 T(St(5)) 5 -4 6 1 3 -3 -6 4 -1 -5 2 Edge Labels: ±1, ±2, ± 3, ± 4, ± 5, ± 6 Vertex Labels: 0, ±1, ±2, ± 3, ± 4 20 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 SEGL for T(St(2n)) SEGL for T(St(10)) Edge Labels: 0, ±1, ±2, ± 3, … , ± 13 Vertex Labels: 0, ±1, ±2, ± 3, …, ± 9 22 Total cycle T(C8) 23 Edge Labels: ±1, ±2, ± 3, … , ± 12 Vertex Labels: ±1, ±2, ± 3, …, ± 8 -4 4 -12 12 3 -5 11 -6 5 -8 -3 8 -1 2 -11 -2 1 10 -7 6 -10 7 9 -9 SEGL of total cycle T(C8) 24 SEGL of total cycle T(Cn) SEGL for T(St(16)) Edge Labels: ±1, ±2, ± 3, … , ± 24 Vertex Labels: ±1, ±2, ± 3, …, ± 16 25 SEGL for T(St(16)) 26 SEGL of total cycle T(Cn)), n ≡ 0 (mod 8) 27 SEGL for the Union of Vertex Disjoint of 3-Cycles 0 3 -1 -4 -3 -2 2 4 1 2 3 1 -2 0 4 -4 3 -1 Edge labels and vertex labels are 0, ±1, ±2, ±3, ±4 28 SEGL for the Union of Vertex Disjoint of 3-Cycles 5 6 1 -5 5 -6 3 2 -1 -3 -5 -6 -4 -3 -1 -2 3 4 1 -4 -2 4 6 2 Edge labels and vertex labels are ±1, ±2, ±3, ±4, ±5, ±6 29 Let a + b + c = 0. -b a c -c -a b 30 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. 31 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 -1 2 3 1 Hence, C4 is not super edge-graceful. 33 Disjoint union of two 4-cycles Edge Labels=Vertex Labels={1, -1, 2, -2, 3, -3, 4, -4} -3 -1 -2 2 1 -4 3 -3 2 3 1 -2 -1 4 4 -4 Hence, the disjoint union of two 4-cycles is SEG. 34 Is the disjoint union of three 4-cycles SEG? Edge Labels=Vertex Labels={±1, ±2, ±3, ±4, ±5, ±6} 35 An Open Problem: The disjoint union of m 4-cycles is super edge-graceful if m>3. 36 37 Thank You 38