AI 中文总结
该研究证明每个2-连通顶立方图可三边着色，为Tutte三边着色猜想提供最终证明，其构造性方法可在O(n²)时间内完成三边着色，且计算过程可通过生成式AI复现。
AI 中文摘要
若图G存在一个顶点v，使得G-v为平面图，则称G为\textit{顶（apex）}图。我们证明每个2-连通顶立方图都是可三边着色的，该结果为1966年著名的Tutte三边着色猜想提供了最终的证明环节。该证明及结果推广了四色定理的证明，四色定理的证明需要计算机验证。与四色定理的前期证明类似，本证明是构造性的。更准确地说，对于含n个顶点的2-连通顶立方图G，我们的可归约性与放电（discharging）过程可在O(n²)时间内得到G的一个三边着色。作为对我们计算机验证的额外可复现性检查，使用生成式AI系统从附录中给出的详细伪代码重构的独立实现，复现了所需的计算结果。这些重构不属于该定理数学证明的组成部分，但为计算的可复现性提供了额外证据。
英文摘要
A graph $G$ is \emph{apex} if $G$ has a vertex $v$ such that $G-v$ is planar. We prove that every $2$-connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks. As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a $2$-connected apex cubic graph $G$ on $n$ vertices, our reducibility and discharging procedure yields a three-edge-coloring of $G$ in $O(n^2)$ time. As an additional reproducibility check for our computer checks, independent implementations reconstructed from the detailed pseudocode (given in the appendix) using generative AI systems reproduced the required computational results. These reconstructions are not part of the mathematical justification of the theorem, but provide additional evidence for the reproducibility of the computations.
CommentsWe provide detailed pseudocode specifying the computer-assisted parts of the proof. Each pseudocode is linked to the corresponding function in the source code in GitHub https://github.com/three-edge-coloring-apex-cubic-graphs
