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Two counterexamples to a conjecture about even cycles

David Conlon, Eion Mulrenin, Cosmin Pohoata

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.

Two counterexamples to a conjecture about even cycles

Abstract

A conjecture of Verstraëte states that for any fixed there exists a positive constant such that any -free graph contains a -free subgraph with at least edges. For , this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for and : the first comes from a recent construction of a dense -free subgraph of the hypercube and the second from Wenger's construction for extremal -free graphs.

Paper Structure

This paper contains 4 sections, 7 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm: hypercube']}
  3. Proof of Theorem \ref{['thm: wenger']}
  4. Concluding remarks

Key Result

Theorem 1.2

For any positive constant $c$ and any positive integer $N_0$, there exists an integer $N \geq N_0$ and a $C_{10}$-free bipartite graph $G$ on $N$ vertices with the property that every subgraph with at least $c|E(G)|$ edges contains a $C_8$.

Theorems & Definitions (12)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Grebennikov and Marciano, 2025 GM25
  • Theorem 2.2: Chung, 1992 Chung
  • proof : Proof of Theorem \ref{['thm: hypercube']}
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 2 more