Chromatic and achromatic numbers of unitary addition Cayley graphs

Let R be a ring. The unitary addition Cayley graph of R, denoted $U\left(R\right)$, 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 $Z_n$, 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 $U\left(Z_n\right)$ in the case that n is the product of two primes. We prove that the achromatic number of $U\left(Z_{3q}\right)$ 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.