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 \{1,2\} such that for any vertex v\in V(G) with‎ ‎f(v)=\emptyset the condition \bigcup_{u\in N(v)}f(u)=\{1,2\}‎ ‎is fulfilled‎, ‎where N(v) is the open neighborhood of v‎. ‎The ‎weight of a 2RDF f is the value \omega(f)=\sum_{v\in V}|f‎ ‎(v)|‎. ‎The 2-rainbow  domination number of a graph G‎, ‎denoted by \gamma_{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 first 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, ‎\gamma_{r2}(T)\le a(T)+1‎.

Bounding the rainbow domination number of a tree in terms of its annihilation number Keywords:

annihilation number , 2-rainbow dominating function , 2-rainbow domination number