AI 中文总结
研究最小度至少为三且围长至少为五的几乎 4 - 连通非平面图含\(P/e\)子图情况，利用此结构结果证明有限无桥不含\((P/e)\)子图的多重图有无处零 4 - 流，扩展并补充相关定理，结合多种定理证明，部分由计算机辅助。
AI 中文摘要
设\(P\)为彼得森图且\(e\in E(P)\)。我们证明每个最小度至少为三且围长至少为五的几乎 4 - 连通非平面图都包含\(P/e\)作为一个子图。利用这一结构结果，我们表明每个有限无桥的不含\((P/e)\)子图的多重图都允许一个无处零 4 - 流。这扩展了王、张和张（2009 年）关于不含通过收缩\(P\)的一个完美匹配的三条边所得到的图的定理，并补充了托马斯和汤姆森（2000 年）关于不含\((P - e)\)子图的图的定理。证明结合了托马斯和汤姆森（2000 年）的围长为五的结构定理与诺林和托马斯（2016 年）的非平面扩展定理。其有限部分由计算机辅助，并在彼得森图、三重图和篮状图以及十二面体的跳跃和面部交叉扩展中验证了所需的子图模型。
英文摘要
Tutte's $4$-flow conjecture asserts that every finite bridgeless graph with no Petersen minor admits a nowhere-zero $4$-flow. Let $P$ be the Petersen graph and let $e\in E(P)$. We prove that every finite bridgeless $(P/e)$-minor-free multigraph admits a nowhere-zero $4$-flow. Since $P-e$ and $P/e$ are the two maximal proper minors of $P$, combining our result with the theorem of Thomas and Thomson for $(P-e)$-minor-free graphs shows that, for every proper minor $R$ of $P$, every finite bridgeless $R$-minor-free graph admits a nowhere-zero $4$-flow. Equivalently, every finite bridgeless graph without such a flow contains every proper minor of $P$. The proof builds on the girth-five structural framework of Thomas and Thomson together with the nonplanar extension theorem of Norin and Thomas.
