For every integer k≥ 1 and every n-vertex graph G, if e(G)>frac(2k+1)(n-1)2, then there exists an integer r≥ 2 such that G contains cycles of every length in the set 2r,2r+2,…,2r+2(k-1). Equivalently, the sharp extremal bound for graphs containing no k consecutive even cycle lengths is conjectured to be frac(2k+1)(n-1)2.

Verstraëte’s conjecture on consecutive even cycle lengths

Verstraëte posed the conjecture in 2016. It predicts the sharp edge threshold forcing $k$ consecutive even cycle lengths in every $n$ -vertex graph.

Known results

$k=1$ is trivial.
$k=2$ : Gao, Li, Ma, and Xie proved the sharp threshold $e(G)\ge \frac{5(n-1)}{2}$ , with characterized $K_5$ -block exceptions (2025).
Verstraëte proved that average degree at least $8k$ forces $k$ consecutive even cycle lengths, a weaker type of bound.
A 2026 paper proves the conjectured bound when $2k+2\le n\le 4k+1$ , with additional partial results.

August 2026 sufficiently-large- $t$ claim

On August 27, 2026, The Erdos--Gallai bound for consecutive even cycle lengths claimed the conjectured sharp bound, including its equality structure, for every sufficiently large number $t$ of consecutive even lengths. This advances the large- $t$ range but does not cover arbitrary $t$ , and the preprint has not been independently verified.

Current status (as of August 2026): The conjecture is claimed for sufficiently large $t$ , but the general cases remain open and that claim is unverified.

Verstraëte’s conjecture on consecutive even cycle lengths

Primary source

Additional references

Progress summary

Solutions 0