We show that matchings avoiding a certain partial pattern are counted by the 3Catalan numbers. We give a characterization of 12312-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns 12312 and 121323 and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or...

Article history: Received 19 August 2011 Available online xxxx Given positive integers k and where 4 divides k and k/2 k−1, we give a minimum -degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, as well as the hypergrap...

We investigate the following question: “Given an intersecting multi-hypergraph on n points, what fraction of edges must be covered by any of the best 2 points?” (Here “best” means that together they cover the most.) We call this M2(n). This is a special case of a question asked by Erdős and Gyárfás [1] (they considered r–wise intersecting and the best t points), and is a generalization of work ...

We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdős and Rényi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory.

A latin transversal in a square matrix of order n is a set of entries, no two in the same row or column, which are pairwise distinct. A longstanding conjecture of Ryser states that every Latin square with odd order has a latin transversal. Some results on the existence of a large partial latin transversal can be found in [11,6,16]. Mainly motivated by Ryser’s conjecture, Erdős and Spencer [8] p...

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of three edges uses three colours. An induced matching of a graph G is a subset I of edges of G such that the graph induced by the endpoints of I is a matching. In this paper, we prove the NP-completeness of strong 4, 5, and 6-edge-colouring and maximum induced matching in some subclasses of subcubic triangle-f...