Packing 3-vertex paths in claw-free graphs and related topics
Alexander Kelmans

TL;DR
This paper investigates the existence of 3-vertex path packings in claw-free graphs, establishing conditions based on graph connectivity, vertex count modulo 3, and other properties, with implications for domination and graph minors.
Contribution
It provides new criteria for the existence of 3-vertex path factors in 3-connected claw-free graphs, linking packing problems with domination and minor theory.
Findings
Existence of L-factors depending on vertex count mod 3
Conditions for G - P to have an L-factor in cubic or 4-connected graphs
Relations between packing problems, domination, and Hadwiger conjecture
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) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G - {x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or 4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 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…
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