Towards obtaining a 3-Decomposition from a perfect Matching

TL;DR

This paper extends the 3-decomposition conjecture to 3-connected star-like cubic graphs by generalizing techniques used for Hamiltonian graphs, contributing to the understanding of graph decompositions.

Contribution

It introduces a new class of star-like graphs and proves they satisfy the 3-decomposition conjecture, expanding known cases beyond Hamiltonian graphs.

Findings

3-connected star-like graphs satisfy the 3-decomposition conjecture.

Extension of techniques from Hamiltonian graphs to star-like graphs.

Provides new insights into graph decomposition structures.

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.