Cohen–Macaulayness of a class of graphs versus the class of their complements

Abstract

Let n≥2 be an integer. The graph G(n) is obtained by letting all the elements of {0,…,n−1} to be the vertices and defining distinct vertices x and y to be adjacent if and only if gcd⁡(x+y,n)=1. In this paper, well-coveredness, Cohen–Macaulayness, vertex-decomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen–Macaulay and non Cohen–Macaulay graphs.