Planarity of a unit graph: Part-I local case

Abstract

The rings considered in this article are commutative with identity \(1 \neq 0\). Recall that the unit graph of a ring \(R\) is a simple undirected graph whose vertex set is the set of all elements of the ring \(R\) and two distinct vertices \(x, y\) are adjacent in this graph if and only if \(x+y \in U(R)\) where \(U(R)\) is the set of unit elements of ring \(R\). We denote this graph by \(U G(R)\). In this article we classified local ring \(R\) such that \(U G(R)\) is planar.