long cycles:

\[lem:disjointlongcycles\] Let $k \geq 1$ and $\ell\geq 3$ be integers and let $G$ be a graph of minimum degree at least $k\ell-1$. Then $G$ contains $k$ vertex-disjoint cycles that are each of length at least $\ell$.

We proceed by induction on $k$. Starting with the base case $k=1$, let $G$ be a graph of minimum degree at least $\ell-1$. Then a maximal path $P$ in $G$ has at least $\ell$ vertices. Since all neighbours of the first vertex of $P$ are on $P$, we know that $G$ has a cycle of length at least $\ell$, as desired. Now let $k>1$ and assume that the lemma holds true for all values smaller than $k$. Let $G$ be a graph of minimum degree at least $k\ell-1$ and let $C$ be a shortest cycle of length at least $\ell$. Suppose for a contradiction that some $v \in V(G)\backslash V(C)$ has at least $\ell+1$ neighbours on $C:= c_1, c_2, \ldots, c_{|C|}$. In particular, we must have $|C|\geq \ell+1$. Without loss of generality, $c_1$ is a neighbour of $v$. Furthermore, $c_i$ is a neighbour of $v$, for some $i\in \left\{\ell-1, \ell, \ldots, |C|-2\right\}$, since otherwise $v$ would have less than $\ell+1$ neighbours on 