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arXiv · 2509.05822

Chromatic numbers with open and nonzero local modular constraints

Daniel Herden·Jonathan Meddaugh·Mark R. Sepanski·William Clark·Adam Kraus·Ellie Matter·Kingsley Michael·Mitchell Minyard·Maricela Ramirez·Kyle Rosengartner·Elyssa Stephens·John Stephens
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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 $\chi_n(G)$ and $\chi_{(n)}(G)$, respectively. In addition, we provide bounds for $\chi_{(n,k)}(G)$, the minimal order of a proper integer coloring of $G$ with open neighborhood sums congruent to $k\mod n$ (when such a coloring exists) as well as precise values for certain families of graphs.

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Daniel Herden, Jonathan Meddaugh, Mark R. Sepanski, William Clark, Adam Kraus, Ellie Matter, Kingsley Michael, Mitchell Minyard, Maricela Ramirez, Kyle Rosengartner, Elyssa Stephens, John Stephens. 2025-09-06. Chromatic numbers with open and nonzero local modular constraints. https://arxiv.org/abs/2509.05822

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