The edge chromatic number of \(\Gamma_{I}(R)\)

Abstract

For a commutative ring \(R\) and an ideal \(I\) of \(R\), the ideal-based zero-divisor graph is the undirected graph \(\Gamma_{I}(R)\) with vertices \(\{x\in R-I: xy\in I~ \text{for some}~ y\in R-I\}\), where distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy\in I\). In this paper, we discuss the nature of the edges of \(\Gamma_{I}(R)\). We also find the edge chromatic number for the graph \(\Gamma_{I}(R)\).