Chromatic numbers with closed local modular constraints
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arXiv
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| Hauptverfasser: | , , , , , , , , , , , |
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| Format: | Preprint |
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2025
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| author | Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Clark, William Kraus, Adam Matter, Ellie Rosengartner, Kyle Stephens, Elyssa Stephens, John Minyard, Mitchell Michael, Kingsley Ramirez, Maricela |
| author_facet | Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Clark, William Kraus, Adam Matter, Ellie Rosengartner, Kyle Stephens, Elyssa Stephens, John Minyard, Mitchell Michael, Kingsley Ramirez, Maricela |
| contents | 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 $χ_{n,k}(G)$ 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 $χ_{n,k}(G)$ are given along with evaluations of $χ_{n,k}(G)$ for some finite and infinite order graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00406 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chromatic numbers with closed local modular constraints Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Clark, William Kraus, Adam Matter, Ellie Rosengartner, Kyle Stephens, Elyssa Stephens, John Minyard, Mitchell Michael, Kingsley Ramirez, Maricela Combinatorics Primary: 05C78, Secondary: 05C25 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 $χ_{n,k}(G)$ 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 $χ_{n,k}(G)$ are given along with evaluations of $χ_{n,k}(G)$ for some finite and infinite order graphs. |
| title | Chromatic numbers with closed local modular constraints |
| topic | Combinatorics Primary: 05C78, Secondary: 05C25 |
| url | https://arxiv.org/abs/2503.00406 |