D/2m$;

6.  $k+2\ell \leq \lfloor (1+\rho^{1/3})n/D\rfloor$;

7.  for all $X\in \mathcal{V}$, all but at most $\rho n$ vertices $x\in X$ satisfy $d_X(x)\geq D-\rho n$.

For technical reasons, we also introduce weak robust subpartitions. We will use this definition and the following result only in Section \[sec:findingcycles\]. A weak robust subpartition differs from a robust partition mainly in that the disjoint subsets need not be a partition of the vertices. Let $k,\ell\in \mathbb{N}_0$ and $0<\rho\leq\nu\leq\tau\leq \eta <1$. Given a graph $G$ on $n$ vertices, we say that $\mathcal{U}$ is a *weak robust subpartition* of $G$ with parameters $\rho,\nu,\tau,\eta,k,\ell$ if the following conditions hold:

1.  $\mathcal{U}= \{U_1,\dots,U_k,Z_1,\dots,Z_\ell\}$ is a collection of disjoint subsets of $V(G)$;

2.  for all $1\leq i \leq k$, $G[U_i]$ is a $(\rho,\nu,\tau)$-robust expander component of $G$;

3.  for all $1\leq j \leq \ell$, there exists a partition $A_j,B_j$ of $Z_j$ such that $G[Z_j]$ is a bipartite $(\rho,\nu,\tau)$-robust expander component with bipartition $A_j,B_j$;

4.  $\delta(G[X])\geq \eta n$ for all $X\in \mathcal{U}$;

5.  for all $1\leq j\leq \ell$, we have $\delta(G[A_j,B_j])\geq \eta n/2$.

\[lem:weakrobustsubp\] Let $k,\ell,D\in \mathbb{N}_0$ and suppose that $0< 1/n \ll \rho \leq \nu\leq \tau\leq \eta \leq \alpha^2/2 < 1$. Suppose that $G$ is a $D$-regular graph on 