We investigate sufficient conditions, and in case that D be an asymmetrical digraph a necessary and sufficient condition for a digraph to have the following property: “In any induced subdigraph H of D, every maximal independent set meets every non-augmentable path”. Also we obtain a necessary and sufficient condition for any orientation of a graph G results a digraph with the above property. Th...

Suppose that n processors are arranged in a ring and can communicate only with their immediate neighbors. We show that any probabilistic algorithm for 3 coloring the ring must take at least 1 2 log n 2 rounds, otherwise the probability that all processors are colored legally is less than 1 2 . A similar time bound holds for selecting a maximal independent set. The bound is tight (up to a consta...

Let k be a positive integer and G = (V (G), E(G)) a graph. A subset S of V (G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k − 1. The maximum cardinality of a k-independent set of G is the k-independence number βk(G). In this paper, we study the properties of graphs for which the k-independence number changes whenever an edge or vertex is ...

Let G be a graph with n vertices, with independence number α, and with with no Kt+1-minor for some t ≥ 5. It is proved that (2α− 1)(2t − 5) ≥ 2n− 5.

In this paper we will define a cardinal invariant corresponding to the independence number for partitions of ω. By using Cohen forcing we will prove that this cardinal invariant is consistently smaller than the continuum.

We determine the independence number of the Kneser graph on line-plane flags in the projective space PG(4, q).

Let $R$ be a commutative ring. In this paper, by using algebraic properties of $R$, we study the Hase digraph of prime ideals of $R$.

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97–108] that there are no 5-connected planar w...

A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number. We answer a question of Topp and Volkmann [5] and prove that if the Cartesian product of two graphs is well-covered, then at least one of them must be well-covered.