Claw-free Graph - Matchings

Matchings

Sumner (1974) and, independently, Las Vergnas (1975) proved that every claw-free connected graph with an even number of vertices has a perfect matching. That is, there exists a set of edges in the graph such that each vertex is an endpoint of exactly one of the matched edges. The special case of this result for line graphs implies that, in any graph with an even number of edges, one can partition the edges into paths of length two. Perfect matchings may be used to provide another characterization of the claw-free graphs: they are exactly the graphs in which every connected induced subgraph of even order has a perfect matching.

Sumner's proof shows, more strongly, that in any connected claw-free graph one can find a pair of adjacent vertices the removal of which leaves the remaining graph connected. To show this, Sumner finds a pair of vertices u and v that are as far apart as possible in the graph, and chooses w to be a neighbor of v that is as far from u as possible; as he shows, neither v nor w can lie on any shortest path from any other node to u, so the removal of v and w leaves the remaining graph connected. Repeatedly removing matched pairs of vertices in this way forms a perfect matching in the given claw-free graph.

The same proof idea holds more generally if u is any vertex, v is any vertex that is maximally far from u, and w is any neighbor of v that is maximally far from u. Further, the removal of v and w from the graph does not change any of the other distances from u. Therefore, the process of forming a matching by finding and removing pairs vw that are maximally far from u may be performed by a single postorder traversal of a breadth first search tree of the graph, rooted at u, in linear time. Chrobak, Naor & Novick (1989) provide an alternative linear-time algorithm based on depth-first search, as well as efficient parallel algorithms for the same problem.

Faudree, Flandrin & Ryjáček (1997) list several related results, including the following: (r − 1)-connected K_1,r-free graphs of even order have perfect matchings for any r ≥ 2; claw-free graphs of odd order with at most one degree-one vertex may be partitioned into an odd cycle and a matching; for any k that is at most half the minimum degree of a claw-free graph in which either k or the number of vertices is even, the graph has a k-factor; and, if a claw-free graph is (2k + 1)-connected, then any k-edge matching can be extended to a perfect matching.