New approach to the k-independence number of a graph

Abstract

Let G = (V,E) be a graph and k ≥ 0 an integer. A k-independent set S ⊆ V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With αk(G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, αk(G) ≥ k+1/[d]+k+1 n, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99-107].