A sufficient condition for large rainbow domination number

Abstract

ABSTRACTLet k be a positive integer, and set . Let G be a graph. A k-rainbow dominating function (or k-RDF) of G is a function f from to such that for a vertex with , the condition is fulfilled, where is the open neighbourhood of v. The weight of k-RDF f of G is the value . The k-rainbow domination number of G, denoted by , is the minimum weight of a k-RDF of G. We focus on two results (here denote the maximum degree of G): (i) If a graph G satisfies , then (proved in Z. Shao, M. Liang, C. Yin, X. Xu, P. Pavlič, and J. Žerovnik, On rainbow domination numbers of graphs, Inform. Sci. 254 (2014), pp. 225–234). (ii) For any graphs G, (proved in D. Meierling, S.M. Sheikholeslami, and L. Volkmann, Nordhaus–Gaddum bounds on the k-rainbow domatic number of a graph, Appl. Math. Lett. 24 (2011), pp. 1758–1761). In this paper, we give a common improvement of (i) and (ii) for the case where , and prove that . Moreover, by partially using the above result, we also obtain a Nordhaus–Gaddum inequality for the k-rainbow domination number and the k-rainbow domination number of ladders.