We determine the maximum number of edges that a claw-free graph can have, when its maximum degree and matching number are bounded. This is a famous problem that has been studied on general graphs, and for which there is a tight bound. The graphs achieving this bound contain in most cases an induced copy of K1,3, the claw, which motivates studying the question on claw-free graphs. Note that on general graphs, if one of the mentioned parameters is not bounded, then there is no upper bound on the number of edges. We show that on claw-free graphs, bounding the matching number is sufficient for obtaining an upper bound on the number of edges. The same is not true for the degree, as a long path is claw-free. We give exact tight formulas for both when only the matching number is bounded and when both parameters are bounded. We also construct claw-free graphs whose edge numbers match the given bounds.
A graph G of even order is ℓ-extendable if it is of order at least 2ℓ + 2, contains a matching of size ℓ, and if every such matching is contained in a perfect matching of G. In this paper, we study the extendability of lexicographic products of graphs. We characterize graphs G and H such that their lexicographic product is not 1-extendable. We also provide several conditions on the graphs G and H under which their lexicographic product is 2-extendable.
A graph is equimatchable if all of its maximal matchings have the same size. Equimatchable graphs are extensively studied in the literature mainly from structural point of view. Here we provide the first family of forbidden subgraphs of equimatchable graphs. Since equimatchable graphs are by definition not hereditary, this task of finding forbidden subgraphs requires the use of structural results from previous works.