Achromatic Index of Unitary Addition Cayley Graph

Abstract

For a positive integer n>1, the unitary addition Cayley Graph G_n= Cay^+ (Z_n,U_n ) is the graph whose vertex set is Z_n and if U_n={a∈Z_n:gcd⁡(a,n)= ┤ ├ 1}, Z_n the integers modulo n then two vertices a,b are adjacent if and only if a+b∈U_n . In this research, we study about the unitary addition Cayley graphs, G_n = Cay^+ (Z_n,U_n ), and to find the lower bound and upper bound of achromatic index of unitary addition Cayley graph where n is even and we improve the bound of achromatic index of graph G_n when n = 2^k, k is the positive integer. Moreover, we found that the unitary addition Cayley graph G_n is the complete bipartite graph K_(2^(k-1),2^(k-1) ) for n= 2^k.