Independent Dominating Sets In Triangle-Free Graphs

The independent domination number of a graph is the smallest cardinality of an independent set that dominates the graph. In this paper we consider the independent domination number of triangle-free graphs. We improve several of the known bounds as a function of the order and minimum degree, thereby answering conjectures of Haviland

Distances between graphs under edge operations

AbstractWe investigate three metrics on the isomorphism classes of graphs derived from elementary edge operations: the edge move, rotation and slide distances. We derive relations between the metrics, and bounds on the distance between arbitrary graphs and between arbitrary trees. We also consider the sensitivity of the metrics to various graph operations

An upper bound for the Ramsey numbers r(K3,G)

AbstractThe 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 ŕed H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K3,G)⩽2q+1 where G has q edges. In other words, any graph on 2q+1 vertices with independence number at most 2 contains every (isolate-free) graph on q edges. This establishes a 1980 conjecture of Harary. The result is best possible as a function of q

Maximum and minimum toughness of graphs of small genus

AbstractA new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus γ(G) is obtained. For γ > 0, the bound is sharp via an infinite class of extremal graphs all of girth 4. For planar graphs, the bound is t(G) > ϰ(G)/2 − 1. For ϰ = 1 this bound is not sharp, but for each ϰ = 3, 4, 5 and any ϵ > 0, infinite families of graphs {G(ϰ, ϵ)} are provided with ϰ(G(ϰ, ϵ)) = ϰ, but t(G(ϰ, ϵ)) < ϰ/2 − 1 + ϵ.Analogous investigations on the torus are carried out, and finally the question of upper bounds is discussed. Several unanswered questions are posed

On the Independent Domination Number of Regular Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs