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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.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C35 Extremal problems in graph theory
GAP
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##### References:
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