Upper Bounds on the Chromatic Index of Linear Hypergraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908583570964480 |
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| author | Murff, Thomas Arsiwalla, Xerxes D. |
| author_facet | Murff, Thomas Arsiwalla, Xerxes D. |
| contents | We address the problem of finding upper bounds on the chromatic index $q(V,E)$ of linear (and loopless) hypergraphs. The first bound we find is defined through a color-preserving group on a proper and minimally edge-colored linear hypergraph, whose orbits serve as a finer partition to the hypergraph's coloring, thereby yielding an upper bound on $q(V,E)$. The next set of theorems in this paper relates to combinatorial properties of hypergraph coloring. Our results suggest a plausible approach to solving the Berge-Füredi conjecture, providing an upper bound on the chromatic index that directly relates $q(V,E)$ and $Δ([(V,E)]_{2}) + 1$. Furthermore, we provide three sufficient conditions for the conjecture to hold within this framework, when involving the Helly property for hypergraphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper Bounds on the Chromatic Index of Linear Hypergraphs Murff, Thomas Arsiwalla, Xerxes D. Combinatorics Group Theory We address the problem of finding upper bounds on the chromatic index $q(V,E)$ of linear (and loopless) hypergraphs. The first bound we find is defined through a color-preserving group on a proper and minimally edge-colored linear hypergraph, whose orbits serve as a finer partition to the hypergraph's coloring, thereby yielding an upper bound on $q(V,E)$. The next set of theorems in this paper relates to combinatorial properties of hypergraph coloring. Our results suggest a plausible approach to solving the Berge-Füredi conjecture, providing an upper bound on the chromatic index that directly relates $q(V,E)$ and $Δ([(V,E)]_{2}) + 1$. Furthermore, we provide three sufficient conditions for the conjecture to hold within this framework, when involving the Helly property for hypergraphs. |
| title | Upper Bounds on the Chromatic Index of Linear Hypergraphs |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2510.07494 |