K-Independence and thek-residue of a graph

Abstract

Favaron, Mahéo, and Saclé proved that the residue of a simple graph G is a lower bound on its independence number α (G). For k ∈ ℕ, a vertex set X in a graph is called k-independent, if the subgraph induced by X has maximum degree less than k. We prove that a generalization of the residue, the k-residue of a graph, yields a lower bound on the k-independence number. The new bound strengthens a bound of Caro and Tuza and improves all known bounds for some graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 241–249, 1999