ON INDEPENDENT GENERALIZED DEGREES AND INDEPENDENCE NUMBERS IN K(1,M)-FREE GRAPHS

In this paper we use independent generalized degree conditions imposed on K(1, m)-free graphs (for an integer m greater-than-or-equal-to 3) to obtain results involving beta(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 beta(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 beta(G) less-than-or-equal-to s, where s is the larger solution to rs (s - 1) = (n - s)(m - 1)(2s - m). We also explore the relationship between beta(G) and delta(G) (the minimum degree) in K(1, m)-free graphs and provide a generalization for degree sums of sets of more than one vertex.