The modular chromatic number of Cartesian product and join of graphs

A vertex coloring [Formula: see text] is defined as, for every [Formula: see text] and [Formula: see text] the sum of the colors of the neighbors of [Formula: see text] and [Formula: see text] that are different in [Formula: see text] The smallest [Formula: see text] for which [Formula: see text] has a modular [Formula: see text]-coloring is the modular chromatic number of [Formula: see text]. In this paper, we calculate the modular chromatic number of [Formula: see text]-regular circulant graphs and [Formula: see text] where [Formula: see text] is a bipartite graph and [Formula: see text] and we give the partial solution to the Problem 1 and it was posed in [R. Rajarajachozhan and R. Sampathkumar, Modular coloring of the Cartesian products [Formula: see text] and [Formula: see text] Discr. Math. Algor. Appl. 9(6) (2017) 1750075]. Also, we prove the Problem 2 partially and it posed in [N. Paramaguru and R. Sampathkumar, Modular colorings of join of two special graphs, Electron. J. Graph Theory Appl. 2(2) (2014) 139–149]. Furthermore, we find [Formula: see text] where [Formula: see text] is either bipartite graph or [Formula: see text]