The Ramsey number r(H, G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph KN in red and blue, it must contain either a ted H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K,, G)< 2qf 1 where G has q edges. In other words, any graph on 2q+ 1 vertices with independence number at most 2 contains every (isolat...

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of simila...

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paperweprovide anupper boundof theHarary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with themaximumHarary index among all graphs with a given number of cut vertices or vertex connectivity or edge connectivity. In ad...

We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on n vertices. The two players take turns choosing edges within Kn, building up a simple graph. The edges must be chosen according to a set of restrictions R. The winner is the last player to choose an edge that does no...

In a linear forest, each component is a path. The linear arboricity ~(G) of a graph G is defined in Harary [8] as the minimum number of linear forests whose union is G. This invariant first arose in a study [i0] of information retrieval in file systems. A quite similar covering invariant which is well known to the linear arboricity is the arboricity of a graph, which is defined as the minimum n...

Harary, F. , 1997 Graph Theory, Narosa Publishing House. Arumugam, S. , Joseph, J. P. 1999 On graphs with equal domination and connected domination numbers, Discrete Mathematics vol. 206, 45-49. Deo, N. 2005 Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India Private Limited. Saoud, M. , Jebran, J. 2009 Finding A Minimum Dominating Set by Transforming Domi...

A linear forest is a in which every connected component path. The arboricity of graph G the minimum number forests covering all edges. In 1980, Akiyama, Exoo, and Harary proposed conjecture, known as Linear Arboricity Conjecture (LAC), stating that Δ-regular has ⌈ Δ + 1 2 ⌉ . 1988, Alon proved LAC holds asymptotically. 1999, list version was raised by An Wu, called List Conjecture. this article...

The aim of this note is to establish new spectral bounds for the harmonic matrix. The Harary matrix of given a connected graph G of order n, say RD(G), is an n-by-n symmetric matrix, such that (RD(G))ij =  1 dij , if i < j, 0 , if i = j, where dij denotes the distance between the vertices i and j [10, 11]. This matrix (originally known as reciprocal distance matrix [11]) is particulary well-k...

Lower bounds on the Ramsey number r(G, H), as a function of the size of the graphs G and H, are determined. In particular, if H is a graph with n lines, lower bounds for r(H) = r(H, H) and r(K,, H) are calculated in terms of n in the first case and m and n in the second case. For m = 3 an upper bound is also determined. These results partially answer a question raised by Harary about the relati...

The Harary-Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph Kn is H(n) = 1 4 ⌊ n 2 ⌋⌊ n− 1 2 ⌋⌊ n− 2 2 ⌋⌊ n− 3 2 ⌋ . Ábrego et al. [3] introduced the notion of shellability of a drawing D of Kn. They proved that if D is s-shellable for some s ≥ b 2 c, then D has at least H(n) crossings. This is the first combinatorial condition on a dr...