We consider the well-studied problem of finding a perfect matching in d-regular bipartite graphs with 2n vertices and m = nd edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes O(m √ n) time, in regular bipartite graphs, a perfect matching is known to be computable in O(m) time. Very recently, the O(m) bound was improved to O(min{m, n 2.5 ln n d }...

A perfect matching in a 4-uniform hypergraph is a subset of b4 c disjoint edges. We prove that if H is a sufficiently large 4-uniform hypergraph on n = 4k vertices such that every vertex belongs to more than ( n−1 3 ) − ( 3n/4 3 ) edges then H contains a perfect matching. This bound is tight and settles a conjecture of Hán, Person and Schacht.

Let G = (V,E) be a weighted undirected graph with vertex set V , edge set E and a weight function d. Thus, d(u, v) denotes the weight of any edge (u, v) ∈ E. A matching M ⊆ E is a collection of edges such that every node in V is incident to at most one edge in M . The matching is perfect if every node in V is incident to exactly one edge in M . The cost of the matching is given by ∑ (u,v)∈M d(u...

We consider the well-studied problem of finding a perfect matching in d-regular bipartite graphs with 2n vertices and m = nd edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes O(m √ n) time, in regular bipartite graphs, a perfect matching is known to be computable in O(m) time. Very recently, the O(m) bound was improved to O(min{m, n 2.5 lnn d })...

In memory of our friend, Ferran Hurtado. Given a set S = {R1, R2, . . . , R2n} of 2n disjoint open regions in the plane, we examine the problem of computing a non-crossing perfect region-matching: a perfect matching on S that is realized by a set of non-crossing line segments, with the segments disjoint from the regions. We study the complexity of this problem, showing that, in general, it is N...

Given a graph G that admits a perfect matching, we investigate the parameter η(G) (originally motivated by computer graphics applications) which is defined as follows. Among all nonnegative edge weight assignments, η(G) is the minimum ratio between (i) the maximum weight of a perfect matching and (ii) the maximum weight of a general matching. In this paper, we determine the exact value of η for...

A graph Γ1 is a matching minor of Γ if some even subdivision of Γ1 is isomorphic to a subgraph Γ2 of Γ, and by deleting the vertices of Γ2 from Γ the left subgraph has a perfect matching. Motivated by the study of Pfaffian graphs (the numbers of perfect matchings of these graphs can be computed in polynomial time), we characterized Abelian Cayley graphs which do not contain a K3,3 matching mino...

Given a bipartite graph G = (U ∪ V , E) such that | U |=| V | and every edge is labelled true or false or both, the perfect matching free subgraph problem is to determine whether or not there exists a subgraph of G containing, for each node u of U , either all the edges labelled true or all the edges labelled false incident to u, and which does not contain a perfect matching. This problem arise...

The perfect matching problem has a randomized NC algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph, ensures that it has a unique minimum weight perfect matching, with a good probability. We derandomize this lemma for K3,3-free and K5-free bipartite graphs, i.e. we give a d...

We show that every cubic bridgeless graph with n vertices has at least 3n/4 − 10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.