In this article, we relate the spectrum of discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties graph: existence perfect matching and Hamiltonian cycle underlying graph. particular, give family spectral obstructions parametrised by potential for to be matchable (i.e., having matching) or cycle. We base our analysis special case preorder introduced in [8], us...

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 ...