On k-domination and minimum degree in graphs

A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbors in S. The k-domination number (gamma k)(G) is the minimum cardinality of a k-dominating set of G. Different upper bounds on (gamma k)(G) are known in terms of the order n and the minimum degree delta of G. In this self-contained article, we present an Erdos-type result, from which some of these bounds follow. In particular, we improve the bound (gamma k)(G) <= (2k - delta - 1)n/(2k - delta) for (delta + 3)/2 <= k <= delta - 1, proved by Chen and Zhou in 1998. Furthermore, we characterize the extremal graphs in the inequality (gamma k)(G) <= kn/(k + 1), if k <= delta, of Cockayne et al. This characterization generalizes that of graphs realizing (gamma 1)(G) = gamma(G) n/2. (c) 2007 Wiley Periodicals, Inc.