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

A Λ-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. A graph is clawfree if it does not have a subgraph isomorphic to K1,3. Our results include the following. Let G be a 3-connected claw-free graph, x ∈ V (G), e = xy ∈ E(G), and L a 3-vertex path in G. Then (c1) if v(G) = 0 mod 3, then G has a Λ-factor containing (avoiding) e, (c2) if v(G) = 1 mod 3, then G − x has a Λ-factor, (c3) if v(G) = 2 mod 3, then G − {x, y} has a Λ-factor, (c4) if v(G) = 0 mod 3 and G is either cubic or 4-connected, then G − L has a Λ-factor, and (c5) if G is cubic and E is a set of three edges in G, then G − E has a Λ-factor if and only if the subgraph induced by E in G is not a claw and not a triangle.