Many Vertex-Disjoint Even Cycles of Fixed Length in a Graph

Abstract

For every integer $k\ge 3$, we determine the maximum number of edges in 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 $\left(\frac{\text{ex}\left(n,C_{2k}\right)}{\epsilon n},\epsilon n\right)$, where $\epsilon >0$ is a constant depending only on $k$. Moreover, we show that the unique extremal graph is obtained from the complete bipartite graph between $V_1$ of size $k\left(t+1\right)-1$ and $V_2$ of size $n-k\left(t+1\right)+1$, by adding all edges within $V_1$ and exactly one additional edge within $V_2$. The question for $k=2$ and $t=o\left(\frac{\text{ex}\left(n,C_{2k}\right)}{n}\right)$ was explored in prior work [25], revealing different extremal structures in these cases. Our result can be viewed as an extension of the theorems by Egawa and Verstraëte, where the focus was on the existence of many vertex-disjoint cycles of the same length without any length constraints.