Domination and Upper Domination of Direct Product Graphs

TL;DR

This paper investigates domination and upper domination numbers in direct product graphs, providing bounds, constructions, and conjectures, especially focusing on unitary Cayley graphs and their properties related to prime factors.

Contribution

It introduces new bounds and constructions for domination numbers in product graphs, extending previous work and proposing conjectures on upper domination numbers.

Findings

Constructed integers with many prime factors satisfying specific domination inequalities.

Provided lower bounds for domination numbers of direct products of complete graphs.

Proposed and proved a conjecture on upper domination numbers for certain multipartite graphs.

Abstract

The unitary Cayley graph of $\mathbb{Z} /n \mathbb{Z}$, denoted $X_{\mathbb{Z} / n \mathbb{Z}}$, has vertices $0,1, \dots, n-1$ with $x$ adjacent to $y$ if $x-y$ is relatively prime to $n$. We present results on the tightness of the known inequality $\gamma(X_{\mathbb{Z} / n \mathbb{Z}})\leq \gamma_t(X_{\mathbb{Z} / n \mathbb{Z}})\leq g(n)$, 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 $\gamma(X_{\mathbb{Z} / n \mathbb{Z}})\leq\gamma_t(X_{\mathbb{Z} / n \mathbb{Z}})\leq g(n)-1$. Extending work of Meki\v{s}, we give lower bounds for the domination numbers of direct products of complete graphs. We also present a simple conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs and prove the conjecture in certain cases. We end with some open problems.