Domination Parameters of the Unitary Cayley Graph of $\mathbb{Z}/n\mathbb{Z}$

Amanda Burcroff

Published 2018-09-13Version 1

The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a family of infinitely many integers $n$ such that $\gamma_t(X_n) \leq g(n) - 2$, where $\gamma_t$ denotes the total domination number and $g$ denotes the Jacobsthal function. We determine the irredundance number, domination number, and lower independence number of certain direct products of complete graphs and give bounds for these parameters for any direct product of complete graphs. We provide upper bounds on the size of irredundant sets in direct products of balanced, complete multipartite graphs which are asymptotically correct for the unitary Cayley graphs of integers with a bounded smallest prime factor.