Clique number and Chromatic number of a graph associated to a Commutative Ring with Unity

TL;DR

This paper investigates the clique and chromatic numbers of graphs derived from commutative rings with unity, providing bounds, counterexamples to Beck's conjecture, and extending previous results in the field.

Contribution

It establishes bounds for clique and chromatic numbers of such graphs for finite products of rings and constructs infinitely many counterexamples to Beck's conjecture.

Findings

Bounds for clique and chromatic numbers in terms of ring factors

Construction of infinitely many counterexamples to Beck's conjecture

Extension of previous results by Beck, Anderson, and Naseer

Abstract

Let R be a commutative ring with unity (not necessarily finite). The ring R consider as a simple graph whose vertices are the elements of R with two distinct vertices x and y are adjacent if xy=0 in R, where 0 is the zero element of R. In 1988 [8], Beck raised the conjecture that the chromatic number and clique number are same in a graph associated to any commutative ring with unity. In 1993 [2], Anderson and Naseer disproved the conjecture by giving a counter example of the conjecture (Note that, till date this is a only one counter example). In this paper, we find the clique number and bounds for chromatic number of a graph associated to any finite product of commutative rings with unity in terms of its factors (In particular, if R is finite which is not a local ring, we obtain the clique number and bounds for chromatic number of a graph associated to R in terms of its local rings). As a consequence of our results, we construct infinitely many counter examples of the above conjecture. Also, some of the main results, proved by Beck in 1988 [8] and Anderson and Naseer in 1993 [2] are consequences of our results.