Chromatic and achromatic numbers of unitary addition Cayley graphs

TL;DR

This paper investigates the chromatic, clique, and achromatic numbers of unitary addition Cayley graphs over finite rings, providing formulas and bounds especially for rings of odd order and specific cases like products of two primes.

Contribution

It derives formulas for clique and chromatic numbers of these graphs over finite commutative rings with odd order, extending known results to new cases including $bZ_n$ with odd $n$ and products of two primes.

Findings

Formula for clique and chromatic numbers when $R$ is a finite commutative ring with odd order.

Exact achromatic number for $bZ_{3q}$ where $q$ is a prime greater than 3.

Lower bounds for the achromatic number when $n$ is a product of two odd primes.

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.