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

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.