Bondy and Vince proved that a graph of minimum degree at least three contains two cycles whose lengths differ by one or two, which was conjectured by Erd\H{ o}s. Gao, Li, Ma and Xie gave an average degree counterpart of Bondy-Vince's result, stating that every $n$-vertex graph with at least $\frac{ 5}{ 2}(n-1)$ edges contains two cycles of consecutive even lengths, unless $4|(n-1)$ and every block of $G$ is a clique $K_5$. This confirms the case $k=2$ of Verstra\"ete's conjecture, which states that every $n$-vertex graph without $k$ cycles of consecutive even lengths has edge number $e(G)\leq\frac{ 1}{ 2}(2k+1)(n-1)$, with equality if and only if every block of $G$ is a clique of order $2k+1$. Sudakov and Verstra\"{ e}te further conjectured that if $G$ is a graph with maximum number of edges that does not contain $k$ cycles of consecutive even lengths, then every block of $G$ is a clique of order at most $2k+1$. In this paper, we prove the case $k=2$ for Sudakov-Verstra\"{ e}te's conjecture, by extending the results of Gao, Li, Ma and Xie.
Categories: math.CO