Domination and upper domination of direct product graphs

Let X-z/nz denote the unitary Cayley graph of Z/nZ. We present results on the tightness of the known inequality gamma (X-z/nz) <= gamma(t)(X-z/nz) <= g(n), where gamma and gamma(t) 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 gamma(X-z/nz) <= gamma(t) (X-z/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. (C) 2018 Elsevier B.V. All rights reserved.