A short proof of the Goldberg-Seymour conjecture
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914867829538816 |
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| author | Chen, Guantao Hao, Yanli Yu, Xingxing Zang, Wenan |
| author_facet | Chen, Guantao Hao, Yanli Yu, Xingxing Zang, Wenan |
| contents | For a multigraph $G$, $χ'(G)$ denotes the chromatic index of $G$, $Δ(G)$ the maximum degree of $G$, and $Γ(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ and } |V(H)| \text{ odd}\right\}$. As a generalization of Vizing's classical coloring result for simple graphs, the Goldberg-Seymour conjecture, posed in the 1970s, states that $χ'(G)=\max\{Δ(G), Γ(G)\}$ or $χ'(G)=\max\{Δ(G) + 1, Γ(G)\}$. Hochbaum, Nishizeki, and Shmoys further conjectured in 1986 that such a coloring can be found in polynomial time. A long proof of the Goldberg-Seymour conjecture was announced in 2019 by Chen, Jing, and Zang, and one case in that proof was eliminated recently by Jing (but the proof is still long); and neither proof has been verified. In this paper, we give a proof of the Goldberg-Seymour conjecture that is significantly shorter and confirm the Hochbaum-Nishizeki-Shmoys conjecture by providing an $O(|V|^5|E|^3)$ time algorithm for finding a $\max\{Δ(G) + 1, Γ(G)\}$-edge-coloring of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A short proof of the Goldberg-Seymour conjecture Chen, Guantao Hao, Yanli Yu, Xingxing Zang, Wenan Combinatorics For a multigraph $G$, $χ'(G)$ denotes the chromatic index of $G$, $Δ(G)$ the maximum degree of $G$, and $Γ(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ and } |V(H)| \text{ odd}\right\}$. As a generalization of Vizing's classical coloring result for simple graphs, the Goldberg-Seymour conjecture, posed in the 1970s, states that $χ'(G)=\max\{Δ(G), Γ(G)\}$ or $χ'(G)=\max\{Δ(G) + 1, Γ(G)\}$. Hochbaum, Nishizeki, and Shmoys further conjectured in 1986 that such a coloring can be found in polynomial time. A long proof of the Goldberg-Seymour conjecture was announced in 2019 by Chen, Jing, and Zang, and one case in that proof was eliminated recently by Jing (but the proof is still long); and neither proof has been verified. In this paper, we give a proof of the Goldberg-Seymour conjecture that is significantly shorter and confirm the Hochbaum-Nishizeki-Shmoys conjecture by providing an $O(|V|^5|E|^3)$ time algorithm for finding a $\max\{Δ(G) + 1, Γ(G)\}$-edge-coloring of $G$. |
| title | A short proof of the Goldberg-Seymour conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.09403 |