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

A Λ-factor   of a graph GG is a spanning subgraph of GG whose every component is a 3-vertex path. Let v(G)v(G) be the number of vertices of GG and γ(G)γ(G) the domination number of GG. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free   if it does not have an induced subgraph isomorphic to a claw. Our results include the following. Let GG be a 3-connected claw-free graph, x∈V(G)x∈V(G), e=xy∈E(G)e=xy∈E(G), and LL a 3-vertex path in GG. Then (a1)(a1) if v(G)≡0mod3v(G)≡0mod3, then GG has a Λ-factor containing (avoiding) ee, (a2)(a2) if v(G)≡1mod3v(G)≡1mod3, then G−xG−x has a Λ-factor, (a3)(a3) if v(G)≡2mod3v(G)≡2mod3, then G−{x,y}G−{x,y} has a Λ-factor, (a4)(a4) if v(G)≡0mod3v(G)≡0mod3 and GG is either cubic or 4-connected, then G−LG−L has a Λ-factor, (a5)(a5) if GG is cubic with v(G)≥6v(G)≥6 and EE is a set of three edges in GG, then G−EG−E has a Λ-factor if and only if the subgraph induced by EE in GG is not a claw and not a triangle, (a6)(a6) if v(G)≡1mod3v(G)≡1mod3, then G−{v,e}G−{v,e} has a Λ-factor for every vertex vv and every edge ee in GG, (a7)(a7) if v(G)≡1mod3v(G)≡1mod3, then there exist a 4-vertex path Π and a claw YY in GG such that G−Π and G−YG−Y have Λ-factors, and (a8)(a8)γ(G)≤⌈v(G)/3⌉γ(G)≤⌈v(G)/3⌉ and if in addition GG is not a cycle and v(G)≡1mod3v(G)≡1mod3, then γ(G)≤⌊v(G)/3⌋γ(G)≤⌊v(G)/3⌋. We also explore the relations between packing problems of a graph and its line graph to obtain some results on different types of packings and discuss relations between Λ-packing and domination problems.