Chromatic numbers with open and nonzero local modular constraints
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arXiv
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| Auteurs principaux: | , , , , , , , , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909774572945408 |
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| author | Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Clark, William Kraus, Adam Matter, Ellie Michael, Kingsley Minyard, Mitchell Ramirez, Maricela Rosengartner, Kyle Stephens, Elyssa Stephens, John |
| author_facet | Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Clark, William Kraus, Adam Matter, Ellie Michael, Kingsley Minyard, Mitchell Ramirez, Maricela Rosengartner, Kyle Stephens, Elyssa Stephens, John |
| contents | 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 $χ_n(G)$ 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 $k\mod n$ (when such a coloring exists) as well as precise values for certain families of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chromatic numbers with open and nonzero local modular constraints Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Clark, William Kraus, Adam Matter, Ellie Michael, Kingsley Minyard, Mitchell Ramirez, Maricela Rosengartner, Kyle Stephens, Elyssa Stephens, John Combinatorics Primary: 05C78, Secondary: 05C25 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 $χ_n(G)$ 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 $k\mod n$ (when such a coloring exists) as well as precise values for certain families of graphs. |
| title | Chromatic numbers with open and nonzero local modular constraints |
| topic | Combinatorics Primary: 05C78, Secondary: 05C25 |
| url | https://arxiv.org/abs/2509.05822 |