Decomposing Claw-free Subcubic Graphs and $4$-Chordal Subcubic Graphs

TL;DR

This paper proves Hoffmann-Ostenhof's Conjecture for two specific classes of subcubic graphs, namely claw-free and 4-chordal subcubic graphs, by decomposing their edge sets into a spanning tree, a matching, and a 2-regular subgraph.

Contribution

The paper establishes the validity of Hoffmann-Ostenhof's Conjecture for claw-free and 4-chordal subcubic graphs, expanding the classes of graphs where the conjecture holds.

Findings

Conjecture holds for claw-free subcubic graphs

Conjecture holds for 4-chordal subcubic graphs

Edge decompositions verified for these classes

Abstract

Hoffmann-Ostenhof's Conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a $2$-regular subgraph. In this paper, we show that the conjecture holds for claw-free subcubic graphs and $4$-chordal subcubic graphs.