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arXiv · 2603.24515

Two counterexamples to a conjecture about even cycles

David Conlon·Eion Mulrenin·Cosmin Pohoata
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Abstract

A conjecture of Verstraëte states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell = 4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

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David Conlon, Eion Mulrenin, Cosmin Pohoata. 2026-03-25. Two counterexamples to a conjecture about even cycles. https://arxiv.org/abs/2603.24515

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