Domination in direct products of complete graphs

Abstract

Let Xn denote the unitary Cayley graph of Z∕nZ. We continue the study of cases in which the inequality γt(Xn)≤g(n) is strict, where γt denotes the total domination number, and g is the arithmetic function known as Jacobsthal’s function. The best that is currently known in this direction is a construction of Burcroff which gives a family of n with arbitrarily many prime factors that satisfy γt(Xn)≤g(n)−2. We present a new interpretation of the problem which allows us to use recent results on the computation of Jacobsthal’s function to construct n with arbitrarily many prime factors that satisfy γt(Xn)≤g(n)−16. We also present new lower bounds on the domination numbers of direct products of complete graphs, which in turn allow us to derive new asymptotic lower bounds on γ(Xn), where γ denotes the domination number. Finally, resolving a question of Defant and Iyer, we completely classify all graphs G=∏i=1tKni satisfying γ(G)=t+2.