Coloring of cozero-divisor graphs of commutative von Neumann regular rings

Abstract

Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by ${\Gamma }^{\prime }(R)$, is a graph with vertices in $W^{\ast }(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^{\ast }(R)$ are adjacent if and only if $a\notin Rb$ and $b\notin Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.

9 pages