Erdős–Ko–Rado for perfect matchings

Lindzey, Nathan

doi:10.1016/j.ejc.2017.05.005

Cited by 18 publications

(28 citation statements)

References 16 publications

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“…Further, they conjectured a version of the EKR theorem holds for tintersecting families of perfect matchings, when 2k 3t + 2. In 2018, Lindzey [12] proved this conjecture for all t, provided that k is sufficiently large relative to t. In this paper we prove the conjecture holds for t = 2 and all k 3.…”

Section: Introductionmentioning

confidence: 59%

“…Our first open question is if these approaches can be generalized to prove a version of the Erdős-Ko-Rado theorem for the family of t-intersecting perfect matchings of the complete graph K 2k , where t > 2? This has been done for k sufficiently large relative to t in [12]. In this work, it is quite remarkable that we were able to find a weighted adjacency matrix for M 2 (2k) for which the ratio bound holds with equality that only uses three of the classes of the association scheme.…”

Section: Further Workmentioning

confidence: 89%

“…This method to prove EKR theorems was first developed by Wilson in 1984 [20]. In 2015, Godsil and Meagher applied this method to the family of all perfect matchings of the complete graph K 2k to find the largest set of intersecting perfect matchings (t = 1) [8]; later in 2017 it was applied to t-intersecting perfect matchings by Lindzey [12]. (6)).…”

Section: Definition 22 ([8]mentioning

confidence: 99%

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The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings

Fallat¹,

Meagher²,

Shirazi³

2021

Algebraic Combinatorics

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A perfect matching in the complete graph on 2k vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect matchings are said to be t-intersecting if they have at least t edges in common. The main result in this paper is an extension of the famous Erdős-Ko-Rado (EKR) theorem [4] to 2-intersecting families of perfect matchings for all values of k. Specifically, for k 3 a set of 2-intersecting perfect matchings in K 2k of maximum size has (2k − 5)(2k − 7) • • • (1) perfect matchings.If equality holds, then F consists of all k-subsets containing a fixed t-subset of {1, 2, . . . , n}.Twenty-three years after the publication of Erdős, Ko and Rado's work, Wilson [20] enhanced their results by giving an algebraic proof of the their result with the exact value of f (k, t) for all k and t. Later in 1997, Ahlswede and Khachatrian [1] found all maximum t-intersecting families of k-subsets for all values of n. In 2011, Ellis, Friedgut, and Pilpel [3] showed that the analog of the EKR theorem holds for tintersecting families of permutations of {1, . . . , n}, when n is sufficiently large relative to t. In 2005, Meagher and Moura [13] proved that a natural version of the EKR Manuscript

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Section: Introductionmentioning

confidence: 59%

Section: Further Workmentioning

confidence: 89%

Section: Definition 22 ([8]mentioning

confidence: 99%

See 1 more Smart Citation

The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings

Fallat¹,

Meagher²,

Shirazi³

2021

Algebraic Combinatorics

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“…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.…”

Section: Moreover Equality Holds If and Only If F Is Trivially Intersectingmentioning

confidence: 97%

Eigenvalues of the matching derangement graph

Wong

2017

J Algebr Comb

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“…[23]). Let l(λ) denote the number of parts of λ n, m i denote the number of parts of λ that equal i, and set z λ := i≥1 i m i m i !.…”

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confidence: 99%

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

Curticapean

Lindzey

Nederlof

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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For even k ∈ N, the matchings connectivity matrix M k is a binary matrix indexed by perfect matchings on k vertices; the entry at (M, M ) is 1 iff M ∪ M forms a single cycle. Cygan et al. (STOC 2013) showed that the rank of M k over Z 2 is Θ( √ 2 k ) and used this to give an O * ((2 + √ 2) pw ) time algorithm for counting Hamiltonian cycles modulo 2 on graphs of pathwidth pw, carrying over to the decision problem via witness isolation. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within M k , which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).We present a new technique for a similar "pattern propagation" when only a black-box lower bound on the asymptotic rank of M k is given; no stronger structural insights such as the existence of large permutation submatrices in M k are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes p) parameterized by pathwidth.To apply this technique, we prove that the rank of M k over the rationals is 4 k /poly(k), using the representation theory of the symmetric group and various insights from algebraic combinatorics. We also show that the rank of M k over Z p is Ω(1.57 k ) for any prime p = 2. IntroductionRank is a fundamental concept in linear algebra and has numerous applications in diverse areas of discrete mathematics and theoretical computer science, such as algebraic complexity [8], communication complexity [27], and extremal combinatorics [30], to name only a few examples. A common phenomenon in these areas is that low rank often helps in proving combinatorial upper bounds or designing algorithms, e.g., through representative sets [7,13,21] or the polynomial method (which ultimately relies on fast rectangular matrix multiplication, enabled through low-rank factorizations of problem-related matrices [37]). In particular, rank has recently found applications in fine-grained complexity (see [1] and the references therein) and the closely related area of parameterized complexity. In the latter, several influential results, such as algorithms for kernelization [21], the longest path problem [31], and connectivity problems parameterized by treewidth [12,11,7], rely crucially on low-rank factorizations.In view of the utility of low rank in proving upper bounds, it is natural to ask, conversely, whether high rank translates into lower bounds. Indeed, examples for this connection can be found in communication complexity [22, Section 1.4] and circuit complexity [16]. In the present paper, we find such applications also in finegrained and parameterized complexity: We develop a technique that allows us to transform rank lower bounds into conditional lower bounds for the problem #HC of counting Hamiltonian cycles. The decision version HC of #HC, which asks for the existenc...

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Erdős–Ko–Rado for perfect matchings

Cited by 18 publications

References 16 publications

The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings

The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings

Eigenvalues of the matching derangement graph

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

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