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

• Authors: Alexander Kelmans
• Venue: Discrete Applied Mathematics
• Year: 2011

A @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 @c(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 does not have an induced subgraph isomorphic to a claw. 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 (a1) if v(G)=0mod3, then G has a @L-factor containing (avoiding) e, (a2) if v(G)=1mod3, then G-x has a @L-factor, (a3) if v(G)=2mod3, then G-{x,y} has a @L-factor, (a4) if v(G)=0mod3 and G is either cubic or 4-connected, then G-L has a @L-factor, (a5) if G is cubic with v(G)=6 and E is a set of three edges in G, then G-E has a @L-factor if and only if the subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G)=1mod3, then G-{v,e} has a @L-factor for every vertex v and every edge e in G, (a7) if v(G)=1mod3, then there exist a 4-vertex path @P and a claw Y in G such that G-@P and G-Y have @L-factors, and (a8)@c(G)@?@?v(G)/3@? and if in addition G is not a cycle and v(G)=1mod3, then @c(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 @L-packing and domination problems.