Packing 3-Vertex Paths in Claw-Free Graphs

TL;DR

This paper investigates the existence of 3-vertex path packings (L-factors) in 3-connected claw-free graphs, providing conditions based on vertex count modulo 3 and graph connectivity.

Contribution

It establishes new conditions for the existence of L-factors in claw-free graphs, including cases involving edge and vertex removals, and characterizes when such packings exist.

Findings

G has an L-factor containing a specific edge if v(G) ≡ 0 mod 3.

Removing a vertex or edge under certain conditions preserves L-factors.

Characterization of L-factors in cubic claw-free graphs with respect to specific edge sets.

Abstract

An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) denote the number of vertices of G. A graph is called claw-free if it does not have a subgraph isomorphic to the graph with 4 vertices and 3 edges having a common vertex. Our results include the following. Let G$ be a 3-connected claw-free graph, x be a vertex, e = xy be an edge, and P be a 3-vertex path in G. Then (c1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (c2) if v(G) = 1 mod 3, then G - x has a L-factor, (c3) if v(G) = 2 mod 3, then G - x -y has an L-factor, (c4) if v(G) = 0 mod 3 and G is either cubic or 4-connected, then G - P has an L-factor, and (c5) if G is cubic and E is a set of three edges in G, then G - E has an L -factor if and only if the subgraph induced by E in G is not a claw and not a triangle. Keywords: claw-free graph, cubic graph, L-packing, L-factor.