Nonplanarity of unit graphs and classification of the toroidal ones

Abstract

The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x C y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal. Algebraic combinatorics is an area of mathematics which employs methods of abstract algebra in various combinatorial contexts and vice versa. Associating a graph to an algebraic structure is a research subject in this area and has attracted considerable attention. The research in this subject aims at exposing the relationship between algebra and graph theory and at advancing the application of one to the other. In fact, there are three major problems in this area: (1) characterization of the resulting graphs, (2) characterization of the algebraic structures with isomorphic graphs, and (3) realization of the connections between the algebraic structures and the corresponding graphs. Beck (1988) introduced the idea of a zero-divisor graph of a commutative ring R with nonzero identity. He defined00.R/ to be the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x yD 0. He was mostly concerned with coloring of 00.R/. Beck conjectured that . R/D!.R/, where . R/ and !.R/ denote, respectively, the chromatic number and the clique number of00.R/. Such graphs are called weakly perfect graphs. This investigation of coloring of a commutative ring was then continued by Anderson and Naseer (1993). They gave a counterexample for the above conjecture of Beck. Anderson and Livingston (1999) proposed a different