Bounding the k-rainbow total domination number

Abstract

Let [k]={1,2,…,k}, where the elements of [k] are called colors. Supposing f is a function which assigns a subset of [k] to each vertex of a graph G, if each vertex which is assigned the empty set has all k colors in its neighborhood, then f is called a k-rainbow dominating function for G. If f satisfies the additional condition that for every v∈V(G) such that f(v)={i} for some i∈[k] there exists u∈NG(v) such that i∈f(u), then f is called a k-rainbow total dominating function. The corresponding invariant γkrt(G), which is the minimum sum of numbers of assigned colors over all vertices of G, is called the k-rainbow total domination number of G. We present bounds on this domination invariant in terms of domination, total domination, and k-rainbow (total) domination. We compute γkrt(Ka,b), and use this result in a number of places. We establish that for k≥3, the following bounds are tight: γ(k−1)rt(G)≤γkrt(G)≤kk−1γ(k−1)rt(G); we prove similar bounds related to rainbow domination and total domination. For a graph G, its ordinary domination number γ(G), and k≥2, we prove that γkrt(G)≥2kk+1γ(G), which presents a generalization of a result for the case k=2 by Goddard and Henning (2018); we then consider some implications of this result. We conclude the paper with Vizing-like conjectures for both rainbow domination and rainbow total domination, and relate the later one with the original Vizing conjecture.