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Consider the family of all finite graphs with maximum degree ∆(G) < d and matching number ν(G) < m. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all the cases when the maximal graph is unique. We also provide a new proof of Gallai's lemma concerning factor critical graphs.
This article provides bounds on the size of a 3-uniform linear hypergraph with restricted matching number and maximum degree. In particular, we show that if a 3-uniform, linear family F has maximum matching size ν and maximum degree ∆ such that ∆ ≥ 23 6 ν(1 + 1 ν−1), then |F| ≤ ∆ν.
We show that for a large family of combinatorial statistics on perfect matchings, the moments can be expressed as a linear combination of double factorials with constant coefficients. This gives a stronger analogous result of Chern, Diaconis, Kane and Rhoades on statistics of set partitions, in which case the moments can be expressed as linear combinations(More)
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