Modular chromatic number of $C_m square P_n$

Abstract

A modular $k!$-coloring‎, ‎$kge 2,$ of a graph $G$ is a coloring of the vertices of $G$ with the elements in $mathbb{Z}_k$ having the property that for every two adjacent vertices of $G,$ the sums of the colors of their neighbors are different in $mathbb{Z}_k.$ The minimum $k$ for which $G$ has a modular $k!$-coloring is the modular chromatic number of $G.$ Except for some special cases‎, ‎modular chromatic number of $C_msquare P_n$ is determined‎.