Chromatic numbers with closed local modular constraints

Generalizing the notion of odd-sum colorings, a Z-labeling of a graph G is called a closed coloring with remainder k n if the closed neighborhood label sum of each vertex is congruent to k n. If such colorings exist, we write χn,k(G) for the minimum number of colors used for a closed coloring with remainder k n such that no neighboring vertices have the same color. General estimates for χn,k(G) are given along with evaluations of χn,k(G) for some finite and infinite order graphs.