Abstract
We confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain $k+1$ vertex-disjoint cycles, then it suffices to delete $2k$ vertices to obtain a forest.
Highlights
We investigate the connection between the maximum number of vertex-disjoint cycles in a graph and the minimum number of vertices whose deletion results in a cycle-free graph, i.e. a forest
A cycle packing of agraph G is a set of vertex disjoint cycles that appear in G as subgraphs
We denote the maximum size of a cycle packing of G by cp(G)
Summary
We investigate the connection between the maximum number of vertex-disjoint cycles in a graph and the minimum number of vertices whose deletion results in a cycle-free graph, i.e. a forest. Erdos and Posa [4] showed that there is a constant c such that for any graph G, fvs(G) c · cp(G) log cp(G), and that this upper-bound is tight for some graphs. The best known bound is that every planar graph G satisfies fvs(G) 3 · cp(G), as proved independently by Chappel et al [2], Chen et al [3], and Ma et al [7]. In his PhD Thesis, Munaro [8] considered the case of subcubic graphs and made significant progress.
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