Chromatic and achromatic numbers of unitary addition Cayley graphs

Let R be a ring. The unitary addition Cayley graph of R, denoted 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. This includes the special case when R is Zₙ, 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( Zₙ ) in the case that n is the product of two primes. We prove that the achromatic number of U ( Z3q) is equal to 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.