Upper Bounds for k-Tuple (Total) Domination Numbers of Regular Graphs

Abstract

Let k be a positive integer. A k-tuple (total) dominating set in a graph G is a subset S of its vertices such that any vertex v in G has at least k vertices inside S among its neighbors and v itself (resp. among its neighbors only). The minimum size of a k-tuple (total) dominating set for G is called the k-tuple (total) domination number of G. It is shown in this article that the k-tuple total domination number of a connected $$(k+1)$$-regular graph with n vertices is at most $$\frac{k^2+k-1}{k^2+k}n$$, unless it is the incident graph of a finite projective plane of order k, where this number is $$\frac{k^2+k}{k^2+k+1}n$$. We also show that the k-tuple domination number of a connected k-regular graph with n vertices is at most $$\frac{k^2-1}{k^2}n$$, unless it is a Moore graph of diameter 2, where this number is $$\frac{k^2}{k^2+1}n$$.