Chromatic and achromatic numbers of unitary addition Cayley graphs
Keenan Calhoun, Yeşim Demiroğlu Karabulut, Vincent Pigno, Craig Timmons
TL;DR
This paper analyzes unitary addition Cayley graphs $\mathcal{U}(R)$ of finite rings, focusing on clique, chromatic, and achromatic numbers. It establishes a general exact formula for the clique and chromatic numbers when $R$ is a product of local rings of odd order, yielding explicit expressions $\omega(\mathcal{U}(R))=\chi(\mathcal{U}(R))= m+\prod_{i=1}^m\frac{|R_i|-|M_i|}{2}$ and, in the $R=\mathbb{Z}_n$ case, $\omega=\chi= m+\phi(n)/2^m$. The paper also determines achromatic numbers in key cases, proving $\chi_a(\mathcal{U}(\mathbb{Z}_{3q}))=(3q+1)/2$ for primes $q>3$ and providing a construction that achieves $(pq+1)/2$ colors for $n=pq$ with primes $3\le p<q$, while giving a lower bound in general. Together, these results extend prior work on even moduli and prime powers and reveal intricate connections between ring structure and graph coloring. The findings have implications for algebraic graph theory and related coloring problems in finite rings.
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.