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arXiv · 2609.20724

Longest cycles intersect linearly in highly connected graphs

Jie Ma·Bo Ning·Ziyuan Zhao
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Abstract

A longstanding conjecture attributed to Smith (1984) asserts that for every $k\ge2$, any two longest cycles in a $k$-connected graph share at least $k$ vertices. In this paper, we prove the first linear lower bound, showing that any two longest cycles in a $k$-connected graph share at least $k/600$ vertices. Departing from previous Turán-type extremal arguments, we develop a novel structural approach that also yields applications to related problems on longest cycles and paths.

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Jie Ma, Bo Ning, Ziyuan Zhao. 2026-09-17. Longest cycles intersect linearly in highly connected graphs. https://arxiv.org/abs/2609.20724

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