BOUNDING THE RAINBOW DOMINATION NUMBER OF A TREE IN TERMS OF ITS ANNIHILATION NUMBER

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set f1; 2g such that for any vertex v 2 V (G) with f(v) = ; the condition Su2N(v) f(u) = f1; 2g is fulبهlled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value !(f) = Pv2V jf(v)j. The 2-rainbow domination number of a graph G, denoted by r2(G), is the minimum weight of a 2RDF of G. The annihilation number a(G) is the largest integer k such that the sum of the rst k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T with at least two vertices, r2(T)  a(T)+1.