A perfect matching in a 3-uniform hypergraph on n = 3k vertices is a subset of n3 disjoint edges. We prove that if H is a 3-uniform hypergraph on n = 3k vertices such that every vertex belongs to at least ( n−1 2 ) − ( 2n/3 2 ) + 1 edges then H contains a perfect matching. We give a construction to show that this result is best possible.

Little [12] showed that, in a certain sense, the only minimal non-Pfaffian bipartite matching covered graph is the brace K3,3. Using a stronger notion of minimality than the one used by Little, we show that every minimal nonPfaffian brick G contains two disjoint odd cycles C1 and C2 such that the subgraph G − V (C1 ∪ C2) has a perfect matching. This implies that the only minimal non-Pfaffian so...

We prove that deciding whether a given input word contains as subsequence every possible permutation of integers {1, 2, . . . , n} is coNP-complete. The coNP-completeness holds even when given the guarantee that the input word contains as subsequences all of length n−1 sequences over the same set of integers. We also show NP-completeness of a related problem of Partially Non-crossing Perfect Ma...

It is shown that for any set of 2n points in general position in the plane there is a non-crossing perfect matching of n straight line segments whose total length is at least 2/π of the maximum possible total length of a (possibly crossing) perfect matching on these points. The constant 2/π is best possible and a non-crossing matching whose length is at least as above can be found in polynomial...

In this lecture, we will describe a strongly polynomial time algorithm for the minimum cost perfect matching problem in a general graph. Using a simple reduction discussed in lecture 9, one can also obtain an algorithm for the maximum weight matching problem. We also note that when discussing perfect matching, without loss of generality, we can assume that all weights/costs are non-negative (wh...

Let G = (V;E) be an undirected graph. Given an odd number k = 2l + 1, a matchingM is said to be k-optimal if it does not admit an augmenting path of length less than or equal to k. We prove jM j jM j(l+1)=(l+2), whereM is a maximum cardinality matching. If M is not already (k+ 2)-optimal, using M , in O(jEj) time we compute a (k+2)-optimal matching. We show that starting with any matching, repe...

In this note, we strengthen the inapproximation bound of O(log n) for the labeled perfect matching problem established in J. Monnot, The Labeled perfect matching in bipartite graphs, Information Processing Letters 96 (2005) 81-88, using a self improving operation in some hard instances. It is interesting to note that this self improving operation does not work for all instances. Moreover, based...

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

We reprove that all the matchings constructed during Edmonds’ weighted perfect matching algorithm are optimal among those of the same cardinality (provided that certain mild restrictions are obeyed on the choices the algorithm makes). We conclude that in order to solve a weighted matching problem it is not needed to solve a weighted perfect matching problem in an auxiliary graph of doubled size...

An even order graph G with a perfect matching is k-extendable if for every matching M of size k in G, there exists a perfect matching in G containing all the edges of M. In this note, we estabUsh a necessary and sufficient condition for a graph to be k-extendable in terms of its independence number. All graphs considered in this note are finite, connected, loopless and have no multiple edges. F...