Extremal trees for new lower bounds on the k-independence number.

Abstract

Let k be a positive integer and G = (V; E) a graph. A subset S of V is a k-independent set of G if the maximum degree of the subgraph induced by the vertices of S is less or equal to k 1. The maximum cardinality of a k-independent set of G is the k-independence number k (G). We give lower bounds on k (G) in terms of the order, the chromatic number and the number of supports vertices. Moreover we characterize extremal trees attaining these bounds.