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Erdős-Ko-Rado for perfect matchings. (English) Zbl 1369.05173
Summary: A perfect matching of a complete graph \(K_{2 n}\) is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if \(\mathcal{F}\) is family of intersecting perfect matchings of \(K_{2 n}\), then \(|\mathcal{F}| \leq(2(n-1)-1)!!\) and if equality holds, then \(\mathcal{F} = \mathcal{F}_{ij}\) where \(\mathcal{F}_{ij}\) is the family of all perfect matchings of \(K_{2 n}\) that contain some fixed edge \(ij\). We give a short algebraic proof of this result, resolving a question of C. D. Godsil and K. Meagher [ibid. 30, No. 2, 404–414 (2009; Zbl 1177.05010)]. Along the way, we show that if a family \(\mathcal{F}\) is non-Hamiltonian, that is, \(m\cup m^\prime \ncong C_{2 n}\) for any \(m\), \(m^\prime \in \mathcal{F}\), then \(|\mathcal{F}| \leq(2(n-1)-1)!!\). Our results make ample use of a symmetric commutative association scheme arising from the Gelfand pair \((S_{2 n}, S_2 \wr S_n)\). We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C35 Extremal problems in graph theory
Full Text: DOI
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