Even cycles in graphs avoiding longer even cycles

David Conlon; Eion Mulrenin; Cosmin Pohoata

Even cycles in graphs avoiding longer even cycles

Combinatorics 2025-01-24 v2

Authors: David Conlon , Eion Mulrenin , Cosmin Pohoata

Abstract

A conjecture of Verstra\"ete 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 K\"uhn and Osthus. We show that $C_6$ and $C_{2k}$ satisfy the conjecture for all odd $k$ , but observe that a recent construction of a dense $C_{10}$ -free subgraph of the hypercube yields a counterexample to the conjecture for $C_8$ and $C_{10}$ .

Keywords

graph cycles hadwiger conjecture graph theory

Cite

@article{arxiv.2501.13036,
  title  = {Even cycles in graphs avoiding longer even cycles},
  author = {David Conlon and Eion Mulrenin and Cosmin Pohoata},
  journal= {arXiv preprint arXiv:2501.13036},
  year   = {2025}
}

Comments

Our main result follows from the work of K\"uhn and Osthus [J. Graph Theory 48 (2005), 147--156]. In particular, our Theorem 1.4 can be proved using their Lemma 10, with g=6

Even cycles in graphs avoiding longer even cycles

Abstract

Keywords

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