Reductions for the 3-Decomposition Conjecture

TL;DR

This paper advances the understanding of the 3-decomposition conjecture for cubic graphs by introducing a new reformulation, identifying reducible configurations, and proving the conjecture for specific graph classes and small graphs.

Contribution

It introduces the HIST-extension conjecture, identifies key reducible configurations, and proves the conjecture for certain graph classes and all graphs up to 20 vertices.

Findings

Certain graphs are reducible configurations for the conjecture.

All graphs of order at most 20 satisfy the 3-decomposition conjecture.

The conjecture holds for 3-connected graphs with bounded tree-width or path-width.

Abstract

The 3-decomposition conjecture is wide open. It asserts that every finite connected cubic graph can be decomposed into a spanning tree, a disjoint union of cycles, and a matching. We show that every such decomposition is derived from a homeomorphically irreducible spanning tree (HIST). This allows us to propose a novel reformulation of the 3-decomposition conjecture: the HIST-extension conjecture. We also prove that the following graphs are reducible configurations with respect to the 3-decomposition conjecture: the triangle, the K_{2,3}, the Petersen graph with one vertex removed, the claw-square, the twin-house, and the domino. As an application, we show that all 3-connected graphs of tree-width at most 3 or of path-width at most 4 satisfy the 3-decomposition conjecture and that a 3-connected minimum counterexample to the conjecture is triangle-free, all cycles of length at most 6 are induced, and every edge is in the centre of an induced P_6. Finally, we automate the naive part of the process of checking whether a configuration is reducible and we prove that all graphs of order at most 20 satisfy the 3-decomposition conjecture.