Planar Graphs with Ore-degree at Most seven is strongly $13$-edge-colorable
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914027727224832 |
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| author | Nelson, Seth Yu, Gexin |
| author_facet | Nelson, Seth Yu, Gexin |
| contents | A strong edge-coloring of a graph $G$ is a coloring of edges of $G$ such that every color class forms an induced matching. The strong chromatic index is the minimum number of colors needed to color the graph. The Ore-degree $θ(G)$ of a graph $G$ is the maximum sum of degrees of adjacent vertices. We show that every planar graph $G$ with $θ(G)\le 7$ has strong chromatic index at most $13$. This settles a conjecture of Chen et al in the planar case. We use a discharging method, and apply Combinatorial Nullstellensatz to show reducible configurations. We provide an algorithm to allow Combinatorial Nullstellansatz extracting coefficients from large polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06808 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar Graphs with Ore-degree at Most seven is strongly $13$-edge-colorable Nelson, Seth Yu, Gexin Combinatorics A strong edge-coloring of a graph $G$ is a coloring of edges of $G$ such that every color class forms an induced matching. The strong chromatic index is the minimum number of colors needed to color the graph. The Ore-degree $θ(G)$ of a graph $G$ is the maximum sum of degrees of adjacent vertices. We show that every planar graph $G$ with $θ(G)\le 7$ has strong chromatic index at most $13$. This settles a conjecture of Chen et al in the planar case. We use a discharging method, and apply Combinatorial Nullstellensatz to show reducible configurations. We provide an algorithm to allow Combinatorial Nullstellansatz extracting coefficients from large polynomials. |
| title | Planar Graphs with Ore-degree at Most seven is strongly $13$-edge-colorable |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.06808 |