A Combinatorial Analysis of Zero-Divisor Graphs on Certain Polynomial Rings

Abstract

The colouring of zero-divisor graphs on certain quotients of polynomial rings can be effectively studied by virtue of a natural correspondence between the algebraic and the combinatorial structure. Accordingly, in the present paper we revisit a nice counterexample involving the chromatic number of zero-divisor graphs, and provide an alternative proof which mostly relies on a purely combinatorial argument. In the same vein, we generalise the counterexample by altering the initial data and studying some chromatic properties of a finite family of zero-divisor graphs.

Key Words:

• Chromatic Number
• Polynomial ring
• Zero-divisor graph

2010 Mathematics Subject Classification:

• 13A99 (05C25; 05C15)

ACKNOWLEDGMENT

The author is grateful to the referees, for their valuable suggestions and remarks.

Notes

Communicated by D. Macpherson.