In this paper we answer a question posed [1]. We show that χ(H2m,3m+2) = m + k + 1 such that k = m/2 and H2m,3m+2 is the Harary graph. Mathematics Subject Classification: 05C15

J. A. Bondy and U. S. R. Murty, Graph theory with Applications, London: MacMillan (1976). R. M. Gera, On Dominator Colorings in Graphs, Graph Theory Notes of New York LIT, 25-30 (2007) F. Harary, Graph Theory, Narosa Publishing 1969. D. Michalak, On Middle and Total graphs with Coarseness Number Equal 1, Lecture Notes in Mathematics, Volume 1018: Graph Theory, Springer-Verlag, Berlin, 139 – 150...

A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterizing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis.

In this paper, we consider Harary graphs as candidate solutions for the design of a physical network topology that achieves a high level of reliability using unreliable network elements. Our network model, which is motivated by the use of all-optical networks for high-reliability applications, is one in which nodes are invulnerable and links are subject to failure in a statistically independent...

Players A and B alternatively colour edges of a graph G, red and blue respectively. Let Lsym(G) be the largest number of moves during which B can keep the red and blue subgraphs isomorphic, no matter how A plays. This function was introduced by Harary, Slany and Verbitsky who in particular showed that for complete bipartite graphs we have Lsym(Km,n) = mn 2 if mn is even and that Lsym(K2m+1,2n+1...

Let G = (V, E) be a graph with v vertices and e edges. An (a, d)-vertex-antimagic total labeling is a bijection λ from V (G) ∪ E(G) to the set of consecutive integers 1, 2, . . . , v + e, such that the weights of the vertices form an arithmetic progression with the initial term a and common difference d. If λ(V (G)) = {1, 2, . . . , v} then we call the labeling a super (a, d)-vertex-antimagic t...

In this note, we show that one of the arguments used by Moazzami for computing the tenacity of the third Harary graph is wrong and then improve the proof. © 2014 Published by Elsevier B.V.

Let G = (V,E) be a graph. A k-coloring of a graph G is a labeling f : V (G) → T , where | T |= k and it is proper if the adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number χ(G) is the least k such that G is k-colorable. Here we study chromatic numbers in some kinds of Harary graphs. Mathematics Subject Classification: 05C15