On independent generalized degrees and independence numbers in K(1, m)-free graphs

Abstract

In this paper we use independent generalized degree conditions imposed on K(1, m)-free graphs (for an integer m⩾3) to obtain results involving β(G), the vertex independence number of G. We determine that in a K(1, m)-free graph G of order n if the cardinality of the neighborhood union of pairs of non-adjacent vertices is a positive fraction of n, then β(G) is bounded and independent of n. In particular, we show that if G is a K(1, m)-free graph of order n such that the cardinality of the neighborhood union of pairs of non-adjacent vertices is at least r, then β(G)⩽s, where s is the larger solution to rs(s−1)=(n−s)(m−1)(2s−m). We also explore the relationship between β(G) and δ(G) (the minimum degree) in K(1, m)-free graphs and provide a generalization for degree sums of sets of more than one vertex.