Domination and upper domination of direct product graphs

Abstract

Let XZ∕nZ denote the unitary Cayley graph of Z∕nZ. We present results on the tightness of the known inequality γ(XZ∕nZ)≤γt(XZ∕nZ)≤g(n), where γ andγt denote the domination number and total domination number, respectively, and g is the arithmetic function known as Jacobsthal’s function. In particular, we construct integers n with arbitrarily many distinct prime factors such that γ(XZ∕nZ)≤γt(XZ∕nZ)≤g(n)−1. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.