Packing 3-vertex Paths In Cubic 3-connected Graphs

TL;DR

This paper investigates the existence of 3-vertex path packings in cubic 3-connected graphs, exploring longstanding conjectures, equivalences, and constructing counterexamples to certain claims.

Contribution

It establishes equivalences among various claims about L-packings in cubic 3-connected graphs and relates these to Reed's dominating graph conjecture, also providing counterexamples.

Findings

Claim (C) is equivalent to several stronger claims.

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

Counterexamples show some claims are best possible.

Abstract

A subgraph (a spanning subgraph) of a graph G whose all components are 3-vertex paths is called an L-packing (respectively, an L-factor} of G. We discuss the following old PROBLEM (A. Kelmans, 1984). Is the following claim true? (C) If G is a cubic 3-connected graph, then G has an L-packing that avoids at most two vertices of G. We show, in particular, that claim (C) is equivalent to some seemingly stronger claims (see Theorem 3.1 below). For example, if G is a cubic 3-connected graph and the number of vertices of G is divisible by three, then then the following claims are equivalent: G has an L-factor, for every edge e of G there is an L-factor of G avoiding (containing) e, G - {e,f} has an L-factor for every two edges e and f of G, and G - P has an L-factor for every 3-vertex path P in G. It follows that if claim (C) is true, then Reed's dominating graph conjecture is true for cubic 3-connected graphs. We also show that certain claims in Theorem 3.1 are best possible. We give a construction providing infinitely many cyclically 6-connected graphs G with two disjoint 3-vertex paths P and P' such that the number of vertices of G is divisible by three and G - P- P' has no L-factor. Keywords: cubic 3-connected graph, 3-vertex path packing, 3-vertex path factor, domination.