Chromatic numbers with open and nonzero local modular constraints

In this paper, we explore chromatic numbers subject to various local modular constraints. For fixed n, we consider proper integer colorings of a graph G for which the closed and open neighborhood sums have nonzero remainders modulo n and provide bounds for the associated chromatic numbers χₙ(G) and χ₍ₙ₎(G), respectively. In addition, we provide bounds for χ(n,k)(G), the minimal order of a proper integer coloring of G with open neighborhood sums congruent to k n (when such a coloring exists) as well as precise values for certain families of graphs.