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This is a Quantitative Study study.

October 12, 2025Open Access

Chromatic numbers with open and nonzero local modular constraints

Key Points

The chromatic numbers associated with graphs are influenced by local modular constraints, showing bounds exist.
For fixed n, bounds for chromatic numbers are determined, including specific cases for graph families.
Proper integer colorings must satisfy nonzero remainders modulo n for closed and open neighborhood sums.
Results show the existence of specific chromatic values under certain local modular conditions and constraints.

Abstract

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 (₍) (G), respectively. In addition, we provide bounds for χ (₍, ₊) (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.

Chromatic numbers with open and nonzero local modular constraints

Key Points

Abstract

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