Partitioning a graph into a cycle and a sparse graph

TL;DR

This paper proves new theorems about finding cycles in graphs that leave a sparse or low-degree subgraph, extending previous work and applying to edge-colouring problems.

Contribution

It introduces two main theorems establishing bounds on the maximum degree of the residual subgraph after removing a cycle, including a tight bound and a bound for k-connected graphs.

Findings

Every graph has a cycle leaving a subgraph with maximum degree at most half its size.

The bound on maximum degree is proven to be optimal.

For k-connected graphs, the residual subgraph's maximum degree is bounded by a fraction plus a small constant.

Abstract

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.