Many vertex-disjoint even cycles of fixed length in a graph

Abstract: For every integer $k \ge 3$, we determine the extremal structure of an $n$-vertex graph with at most $t$ vertex-disjoint copies of $C_{2k}$ when $n$ is sufficiently large and $t$ lies in the interval $\left[\frac{\mathrm{ex}(n,C_{2k})}{\varepsilon n}, \varepsilon n\right]$, where $\varepsilon>0$ is a constant depending only on $k$. The question for $k = 2$ and $t = o\left(\frac{\mathrm{ex}(n,C_{2k})}{n}\right)$ was explored in prior work~\cite{HHLLYZ23a}, revealing different extremal structures in these cases. Our result can be viewed as an extension of the theorems by Egawa~\cite{Ega96} and Verstra\"{e}te~\cite{Ver03}, where the focus was on the existence of many vertex-disjoint cycles of the same length without any length constraints.