Beck’s conjecture and multiplicative lattices

In this paper, we introduce the multiplicative zero-divisor graph of a multiplicative lattice and study Beck-like coloring of such graphs. Further, it is proved that for such graphs, the chromatic number and the clique number need not be equal. On the other hand, if a multiplicative lattice LL is reduced, then the chromatic number and the clique number of the multiplicative zero-divisor graph of LL are equal, which extends the result of Behboodi and Rakeei (2011) and Aalipour et al. (2012).