发表机构
KU Leuven; Ghent University; Leiden University; Comenius University; University of Modena and Reggio Emilia（荷语鲁汶大学; 根特大学; 莱顿大学; 布拉迪斯拉发康斯坦丁哲学者大学; 摩德纳和雷焦艾米利亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Putman提出的112顶点彼得森着色猜想反例，本文给出不依赖大规模SAT计算的人类可验证证明，简化矛盾为彼得森图线图的结构性质，深化了对构造小装置的理解。
AI 中文摘要
Jaeger提出的彼得森着色猜想指出，每个无桥三次图都存在彼得森着色。最近，Putman给出了一个112顶点的明确反例，并通过SAT求解器验证其不可着色性，该求解器处理的实例包含3640个变量和68324个条款，结果显示其不可满足。我们给出一个简短的人类可验证证明，确认该图确实是反例。我们的证明通过对构造中所用多极子的着色行为进行小型明确有限案例分析来确定，并将最终矛盾简化为彼得森图的线图的一个简单结构性质。除了提供不依赖大规模SAT计算的证明外，我们的方法还能更深入地理解该构造所基于的小装置。
英文摘要
In 1988, Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring; that is, a map $E(G) \to E(P)$ mapping any two adjacent edges of $G$ to two adjacent edges of the Petersen graph $P$. A positive resolution of Jaeger's conjecture would have immediately resolved several other famous and long-standing problems in graph theory. In July 2026, a 68-vertex counterexample was announced on X. Shortly afterwards, Putman independently presented two non-isomorphic 112-vertex counterexamples, relying solely on computer-assisted verification. In this paper, we present two counterexamples of order $52$, currently the smallest known, and provide a purely theoretical proof. In the second part, we construct an infinite family of cyclically $4$-edge-connected cubic graphs without a Petersen coloring for every even order at least $60$. Additionally, through computational verification, we show that any counterexample must have order at least $40$. Moreover, we show that our counterexamples provide a negative answer to other related problems. Finally, we conclude the paper by discussing key open problems and highlighting avenues for future work.
