arXiv 2407.094032024-07-12math.CO

Goldberg-Seymour猜想的一个简短证明

A short proof of the Goldberg-Seymour conjecture

Guantao Chen, Yanli Hao, Xingxing Yu, Wenan Zang

AI总结：

本文给出了Goldberg-Seymour猜想的简短证明，并提出时间复杂度为$O(|V|^5|E|^3)$的多项式算法，证实了Hochbaum-Nishizeki-Shmoys关于边着色可多项式时间求解的猜想。

摘要：

对于多重图$G$，$\chi'(G)$表示$G$的边色数，$\Delta(G)$表示$G$的最大度，且$\Gamma(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ 且 } |V(H)| \text{ 为奇数}\right\}$。作为Vizing关于简单图的经典着色结果的推广，20世纪70年代提出的Goldberg-Seymour猜想指出，$\chi'(G)=\max\{\Delta(G), \Gamma(G)\}$或$\chi'(G)=\max\{\Delta(G) + 1, \Gamma(G)\}$。Hochbaum、Nishizeki和Shmoys于1986年进一步猜想，这种着色可以在多项式时间内找到。Chen、Jing和Zang于2019年宣布了Goldberg-Seymour猜想的一个长篇证明，Jing最近消除了该证明中的一种情况（但证明仍然很长）；且这两个证明都尚未得到验证。在本文中，我们给出了Goldberg-Seymour猜想的一个明显更短的证明，并通过提供一种时间复杂度为$O(|V|^5|E|^3)$的算法来寻找$G$的$\max\{\Delta(G) + 1, \Gamma(G)\}$-边着色，从而证实了Hochbaum-Nishizeki-Shmoys猜想。

英文摘要：

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 $G$.

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