A characterization of some types of Cayley graphs and addition Cayley graphs and their total chromatic numbers

Let Z(n) be the set of integer modulo n. The Cayley graph on Z(n) is an undirected graph whose vertex set is. n and two vertices a, b are adjacent if and only if a - b is an element of S subset of Z(n) \{0}. The addition Cayley graph on Z(n) is a graph whose vertex set is Z(n) and two vertices a, b are adjacent if and only if. a + b is an element of A subset of Z(n) . In this paper, we characterize Cayley graphs and addition Cayley graphs of even orders. Their basic properties of them are investigated. We also give exact values for the total chromatic numbers of Cayley graphs and addition Cayley graphs where their orders are even integers. Moreover, we provide examples to illustrate these results.