Three-edge-coloring apex cubic graphs

2026-08-24

AI Summary

本文证明了每个2-连通尖点三次图都是三边可着色的，通过一种减少和放电程序在O(n^2)时间内构造性地解决了这个问题。

Abstract

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.

Problem

• three-edge-coloring
• apex cubic graph
• Tutte's conjecture

Innovation

• three-edge-coloring
• apex cubic graphs
• reducibility and discharging procedure
• generative AI systems