Nowhere-zero 4-flows in graphs excluding a proper minor of the Petersen graph

Abstract

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.

Disclosure

“, or reporting of the research. Declaration of competing interest. The author declares no known competing financial interests or personal relation- ships that could have appeared to influence the work reported in this paper. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process. During the preparation of this work, the author used OpenAI’s ChatGPT and Anthropic’s Claude for brainstorming, literature search, LATEX drafting, and editorial assistance”

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Drafting limited passages

Multiplier

5

Verified

2026-08-23T19:08:29.454533Z

Structural counts