Packing 3-vertex paths in cubic 3-connected graphs

TL;DR

This paper investigates whether every 3-connected cubic graph can be decomposed into the maximum possible number of disjoint 3-vertex paths, linking this to Reed's dominating graph conjecture.

Contribution

It establishes the equivalence of the main claim with stronger conjectures and connects it to Reed's dominating graph conjecture for cubic 3-connected graphs.

Findings

Claim (P) is equivalent to some stronger claims.

If claim (P) holds, Reed's conjecture is true for these graphs.

The paper discusses the longstanding problem in graph theory.

Abstract

Let v(G) and p(G) be the number of vertices and the maximum number of disjoint 3-vertex paths in G, respectively. We discuss the following old Problem: Is the following claim (P) true ? (P) if G is a 3-connected and cubic graph, then p(G) = [v(G)/3], where [v(G)/3] is the floor of v(G)/3. We show, in particular, that claim (P) is equivalent to some seemingly stronger claims. It follows that if claim (P) is true, then Reed's dominating graph conjecture (see [14]) is true for cubic 3-connected graphs.