Towards obtaining a 3-Decomposition from a perfect Matching

Abstract

A decomposition of a graph is a set of subgraphs whose edges partition those of G. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list.In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching M from a cubic graph G leaves a disjoint union of cycles. Contracting these cycles yields a new graph G M . The graph G is star-like if G M is a star for some perfect matching M , making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.

Reductions for the 3-Decomposition Conjecture

“…If a cubic graph G decomposes into a spanning tree T , a 2-regular graph C, and a matching M , then (T, C, M ) is a 3-decomposition of G. Until the present day, the 3-decomposition conjecture remains wide open. The conjecture has been proved for a variety of subclasses of cubic graphs, for example planar cubic graphs [HOKO18], 3-connected cubic graphs of tree-width 3 [Hei19], traceable cubic graphs [AAHM16] (a graph is traceable if it admits a Hamiltonian path), and generalised Hamiltonian cubic graphs, see [BK20] and [XZZ20]. All of these results exploit one of the following two approaches:…”