Chromatic numbers with closed local modular constraints

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