Partitioning a graph into a cycle and a sparse graph

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 $\Delta(G[V(G)\setminus V©])\leq \frac12(|V(G)\setminus V©|-1)$\mathrm{.}Theboundonthemaximumdegree$\Delta(G[V(G)\setminus V©])$ is best possible. The second theorem is that every $k$-connected graph $G$ has a cycle with $\Delta(G[V(G)\setminus V©])\leq \frac1{k+1}|V(G)\setminus V©|+3$. We also give an application of this second theorem to a conjecture about partitioning edge-coloured complete graphs into monochromatic cycles.