Bounding the k-rainbow total domination number

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 fis called a k-rainbow dominating function for G. If f satisfies the additional condition that for every v is an element of V(G) such that f( v) ={i} for some i is an element of[k] there exists u is an element of N-G(v) such that i is an element of f(u), then fis called a k-rainbow total dominating function. The corresponding invariant gamma(krt)(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 gamma(krt)(K-a,K-b), and use this result in a number of places. We establish that for k = 3, the following bounds are tight: gamma((k-1)rt)(G) <= gamma(krt)(G) <= k/k-1 gamma((k-1)rt)(G); we prove similar bounds related to rainbow domination and total domination. For a graph G, its ordinary domination number gamma(G), and k >= 2, we prove that gamma(krt)(G) >= 2k/k+1 gamma(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. (C) 2021 Elsevier B.V. All rights reserved.