say that $G[U]$ is a *$(\rho,\nu,\tau)$-robust expander component* of $G$ if $U$ is a $\rho$-component and $G[U]$ is a robust $(\nu,\tau)$-expander. We say that $G[U]$ is a *bipartite $(\rho,\nu,\tau)$-robust expander component* with bipartition $A,B$ if $U$ is $\rho$-close to bipartite with bipartition $A,B$ and $G[U]$ is a bipartite robust $(\nu,\tau)$-expander with bipartition $A,B$.

We now introduce the concept of a robust partition, which is central to our result.

[**Robust partitions**]{} - Let $k,\ell, D \in \mathbb{N}$ and $0<\rho\leq \nu\leq \tau<1$. Given an $n$-vertex, $D$-regular graph $G$, we say that $\mathcal{V}$ is a *robust partition of $G$ with parameters $\rho,\nu,\tau,k,\ell$* if the following hold:

1.  $\mathcal{V} = \{V_1,\dots,V_k,W_1,\dots,W_\ell\}$ is a partition of $V(G)$;

2.  for all $1 \leq i \leq k$, $G[V_i]$ 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 $W_j$ such that $G[W_j]$ is a bipartite $(\rho,\nu,\tau)$-robust expander component with bipartition $A_j,B_j$;

4.  for all $X,X'\in \mathcal{V}$ and all $x\in X$, we have $d_X(x)\geq d_{X'}(x)$; in particular, $d_X(x)\geq D/m$, where $m:= k+\ell$;

5.  for all $1\leq j \leq \ell$, we have $d_{B_j}(u)\geq d_{A_j}(u)$ for all $u\in A_j$ and $d_{A_j}(v)\geq d_{B_j}(v)$ for all $v\in B_j$; in particular, $\delta(G[A_j,B_j])\geq 