  In this paper we investigate results of the form "every graph $G$ has a cycle
$C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small
maximum degree." Such results haven't been studied before, but are motivated by
the Bessy and Thomass\'e Theorem which states that the vertices of any graph
$G$ can be covered by a cycle $C_1$ in $G$ and disjoint cycle $C_2$ in the
complement of $G$.
  There are two main theorems in this paper. The first is that every graph has
a cycle with $\Delta(G[V(G)\setminus V(C)])\leq \frac12(|V(G)\setminus
V(C)|-1)$. The bound on the maximum degree $\Delta(G[V(G)\setminus V(C)])$ is
best possible. The second theorem is that every $k$-connected graph $G$ has a
cycle with $\Delta(G[V(G)\setminus V(C)])\leq \frac1{k+1}|V(G)\setminus
V(C)|+3$. We also give an application of this second theorem to a conjecture
about partitioning edge-coloured complete graphs into monochromatic cycles.
