2024/07/15 by Calhoun, Keenan, Karabulut, Yeşim Demiroğlu, Pigno, Vincent +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.10364
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 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.