Chromatic and achromatic numbers of unitary addition Cayley graphs

Abstract

Let $R$ be a ring. The unitary addition Cayley graph of $R$, denoted $\mathcal{U}(R)$, is the graph with vertex $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is a unit. We determine a formula for the clique number and chromatic number of such graphs when $R$ is a finite commutative ring with an odd number of elements. This includes the special case when $R$ is $\mathbb{Z}_n$, the integers modulo $n$, where these parameters had been found under the assumption that $n$ is even, or $n$ is a power of an odd prime. Additionally, we study the achromatic number of $\mathcal{U}( \mathbb{Z}_n )$ in the case that $n$ is the product of two primes. We prove that the achromatic number of $\mathcal{U} ( \mathbb{Z}_{3q})$ is equal to $\frac{3q+1}{2}$ when $q > 3$ is a prime. We also prove a lower bound that applies when $n = pq$ where $p$ and $q$ are distinct odd primes.