Longest cycles intersect linearly in highly connected graphs
Jie Ma, Bo Ning, Ziyuan Zhao
Abstract
A longstanding conjecture attributed to Smith (1984) asserts that for every k2, 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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