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 ${\bf Z}_4[x,y,w,z]$. As the 90 quotients vary, we obtain a total of 7 graphs, corresponding to 7 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 ${\bf Z}_4[x,y,z]$ -- therefore, the just smaller case -- in order to get a deeper knowledge of the classical counterexample to Beck's conjecture.

Utilizzando la tecnica di decomposizione introdotta in un precedente lavoro, si costruiscono nuovi esempi di grafi di zero-divisori su un anello di polinomi.