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 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 ( Z n ) in the case that n is the product of two primes. We prove that the achromatic number of U ( Z 3 q ) is equal to 3 q + 1 2 when q > 3 is a prime. We also prove a lower bound that applies when n = p q where p and q are distinct odd primes.