A New Zero-divisor Graph Contradicting Beck's Conjecture, and the Classification for a Family of Polynomial Quotients.

We classify all possible zero-divisor graphs of a particular family of quotients of $$\mathbf{Z}_4[x,y,w,z]$$ . As the 90 quotients vary, we obtain a total of 7 graphs, corresponding to seven isomorphism classes, and one of these graphs provides a new example which contradicts Beck's conjecture on the chromatic number of a zero-divisor graph. The algebraic analysis is strongly supported by the combinatorial setting, as already shown in a previous paper, where the graph-theoretical tools were presented and successfully applied to $$\mathbf{Z}_4[x,y,z]$$ -therefore, the just smaller case-in order to get a deeper knowledge of the classical counterexample to Beck's conjecture. [ABSTRACT FROM AUTHOR]