Partitioning a graph into a cycle and a sparse graph

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)∖V(C) has small maximum degree.” Such results haven't been studied before, but are motivated by the Bessy and Thomassé Theorem which states that the vertices of any graph G can be covered by a cycle C1 in G and disjoint cycle C2 in the complement of G.There are two main theorems in this paper. The first is that every graph has a cycle with Δ(G[V(G)∖V(C)])≤12(|V(G)∖V(C)|−1). The bound on the maximum degree Δ(G[V(G)∖V(C)]) is best possible. The second theorem is that every k-connected graph G has a cycle with Δ(G[V(G)∖V(C)])≤1k+1|V(G)∖V(C)|+3. We also give an application of this second theorem to a conjecture about partitioning edge-coloured complete graphs into monochromatic cycles.