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arXiv · 2608.30353

An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision

Douglas M. Chen
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Abstract

Smith's conjecture asserts that in every $k$-connected graph with $k\geq 2$, any two longest cycles intersect in at least $k$ vertices. In this work, we establish an $Ω(k^{8/11})$ bound for this conjecture, improving upon the $Ω(k^{2/3})$ bound of Ma and Zhao. Our proof combines a Ramsey theoretic refinement of the traditional Turán-type approach with computer search.

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Douglas M. Chen. 2026-08-31. An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision. https://arxiv.org/abs/2608.30353

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