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 (Z₃ₐ) is equal to 3q+12 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.