Combinatorial Proof of the Log-Concavity of the Sequence of Matching Numbers

Fork⩾lwe construct an injection from the set of pairs of matchings in a given graphGof sizesl−1 andk+1 into the set of pairs of matchings inGof sizeslandk. This provides a combinatorial proof of the log-concavity of the sequence of matching numbers of a graph. Besides, this injection implies that a certain weighted version of the matching numbers is stronglyx-log-concave in the sense of Sagan (Discrete Math.99(1992), 289–306).