An algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings

Abstract

In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties of the perfect matching polytope. We also prove that the perfect matching derangement graph is not a Cayley graph. * Research supported by NSERC. the natural version of the Erdős-Ko-Rado (EKR) theorem holds for perfect matchings. This theorem shows that the largest set of perfect matchings, with the property that any two intersect, is the set of the all perfect matchings that contain a specific edge.The algebraic method given in this paper is similar to the proof given in [7] that the natural version of the EKR theorem holds for permutations. In this paper we determine the least eigenvalue for the perfect matching derangement graph. This, with the Delsarte-Hoffman bound, implies that a maximum intersecting set of perfect matchings corresponds to a facet in the perfect matching polytope. The characterization of the maximum set of intersecting perfect matchings follows from the characterization of the facets of this polytope. Further, with this characterization, we are able to prove that the perfect matching derangement graph is not a Cayley graph.Meagher and Moura [10] proved a version of the EKR theorem holds for intersecting uniform partitions using a counting argument [10]. This result includes the EKR theorem for perfect matchings. It is interesting that the counting argument in [10] is straight-forward, except for the case of perfect matchings; in this case a more difficult form of the counting method is necessary.

On the intersection density of the symmetric group acting on uniform subsets of small size

“…Note that in contrast to the previous subsection, (t 0 , s 0 ) does not belong to the polytope given in (12). Fortunately, we can prove that these weights still work.…”

Erdős-Ko-Rado results for the general linear group, the special linear group and the affine general linear group

“…For any group G, the derangement graph Γ G is the union of graphs in the conjugacy class association scheme; details can be found in [18]. The vertex set for this association scheme is G, hence it has |G| vertices, and has rank equal to the number of conjugacy classes of G. If C 1 , C 2 , .…”

Eigenvalues of the matching derangement graph

“…Recently, some algebraic proofs of this result were found by Godsil and Meagher [5], and independently by Lindzey [7]. We define the matching derangement graph to be the graph D 2n whose vertex set is M 2n such that two vertices M, M are adjacent if and only if M ∩ M = ∅, i.e.…”

“…The algebraic proofs of Theorem 1.1 are based on the Delsarte-Hoffman bound [5,7]. The least eigenvalue of D 2n was found by Godsil and Meagher [5].…”