Chromatic numbers with closed local modular constraints

Daniel Herden; Jonathan Meddaugh; Mark R. Sepanski; William Clark; Adam Kraus; Ellie Matter; Kyle Rosengartner; Elyssa Stephens; John Stephens; Mitchell Minyard; Kingsley Michael; Maricela Ramirez

arXiv · 2503.00406

Chromatic numbers with closed local modular constraints

Daniel Herden·Jonathan Meddaugh·Mark R. Sepanski·William Clark·Adam Kraus·Ellie Matter·Kyle Rosengartner·Elyssa Stephens·John Stephens·Mitchell Minyard·Kingsley Michael·Maricela Ramirez

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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.

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Daniel Herden, Jonathan Meddaugh, Mark R. Sepanski, William Clark, Adam Kraus, Ellie Matter, Kyle Rosengartner, Elyssa Stephens, John Stephens, Mitchell Minyard, Kingsley Michael, Maricela Ramirez. 2025-03-01. Chromatic numbers with closed local modular constraints. https://arxiv.org/abs/2503.00406

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