On Orthogonal Symmetric Chain Decompositions

Däubel, Karl; Jäger, Sven; Mütze, Torsten; Scheucher, Manfred

doi:10.37236/8531

Cited by 3 publications

(6 citation statements)

References 38 publications

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“…Similarly, by swapping × with its CCW next , rank (ρ) increases by 1. Further by (8), f mod (ρ) = 1 + Σ t j=1 T ρ j − rank (ρ)( mod t + 1) decreases by 1.…”

Section: Proofs Omitted In Sectionmentioning

confidence: 99%

“…For H (8,3), the program returns many solutions. For H(6, 2), the program runs in less than five seconds and finds no solution.…”

Section: A Proofs Omitted In Sectionmentioning

confidence: 99%

“…By subtraction, rank (ρ) − Σ t j=1 O ρ j = 1 + Σ t j=1 T ρ j − rank (ρ)( mod t + 1). Therefore, the two definitions of f mod given in (7) and (8) are equivalent.…”

Section: Proofs Omitted In Sectionmentioning

confidence: 99%

“…Recently, Spink [27] found out three orthogonal chain decompositions of the subset lattice, which implies three disjoint 1-factors of H(v, t). More recently, four orthogonal chain decompositions can be found for v ≥ 60 [8].…”

Section: Proofs Omitted In Sectionmentioning

confidence: 99%

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On 1-Factorizations of Bipartite Kneser Graphs

Jin

2019

Lecture Notes in Computer Science

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It is a challenging open problem to construct an explicit 1factorization of the bipartite Kneser graph H(v, t), which contains as vertices all t-element and (v − t)-element subsets of [v] := {1, . . . , v} and an edge between any two vertices when one is a subset of the other. In this paper, we propose a new framework for designing such 1-factorizations, by which we solve a nontrivial case where t = 2 and v is an odd prime power. We also revisit two classic constructions for the case v = 2t + 1 -the lexical factorization and modular factorization. We provide their simplified definitions and study their inner structures. As a result, an optimal algorithm is designed for computing the lexical factorizations.(An analogous algorithm for the modular factorization is trivial.)On 1-factorizations of Bipartite Kneser Graphs 3 PreliminariesThe subset lattice is the family of all subsets of [v], partially ordered by inclusion. Let P t denote the t-th layer of this subset lattice, whose members are the telement subsets of [v]. Throughout the paper, denote d = v − 2t. Let the words clockwise and counterclockwise be abbreviated as CW and CCW respectively. A representation of the edges of H(v, t). We identify each edge (A, A ) of H(v, t) by a permutation ρ of t 's, t 's, and d ×'s: the (positions of) t ' 's indicate the t elements in A; the t ' 's indicate the t elements that are not in A (recall that A has v − t elements); and the '×'s indicate those in A − A. We do not distinguish the edges with their corresponding permutations.Denote [t , t , d×] as the multiset of 2t + d characters with t ' 's, t ' 's, and d '×'s. Giving a 1-factorization of H(v, t) is equivalent to giving a labeling function f from the 2t+d t,t,d permutations of [t , t , d×] to 1, . . . , t+d d so that(a) f (ρ) = f (σ) for those pairs ρ, σ who admit the same positions for t 's; and (b) f (ρ) = f (σ) for those pairs ρ, σ who admit the same positions for t 's.

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“…Similarly, by swapping × with its CCW next , rank (ρ) increases by 1. Further by (8), f mod (ρ) = 1 + Σ t j=1 T ρ j − rank (ρ)( mod t + 1) decreases by 1.…”

Section: Proofs Omitted In Sectionmentioning

confidence: 99%

“…For H (8,3), the program returns many solutions. For H(6, 2), the program runs in less than five seconds and finds no solution.…”

Section: A Proofs Omitted In Sectionmentioning

confidence: 99%

“…By subtraction, rank (ρ) − Σ t j=1 O ρ j = 1 + Σ t j=1 T ρ j − rank (ρ)( mod t + 1). Therefore, the two definitions of f mod given in (7) and (8) are equivalent.…”

Section: Proofs Omitted In Sectionmentioning

confidence: 99%

Section: Proofs Omitted In Sectionmentioning

confidence: 99%

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On 1-Factorizations of Bipartite Kneser Graphs

Jin

2019

Lecture Notes in Computer Science

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“…While the maximum‐sized antichain is more or less unique (if n is odd, there are two maximum antichains, otherwise it is unique), there are many different ways to partition

2^{[n]}

into the minimum number of chains. In general, chain decompositions of the Boolean lattice into the minimum number of chains are extensively studied, see, for example, [8, 11, 14–16, 20, 21, 34, 35].…”

Section: Introductionmentioning

confidence: 99%

Uniform chain decompositions and applications

Sudakov

Tomon

Wagner

2021

Random Struct Algorithms

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The Boolean lattice 2[n] is the family of all subsets of [n]={1,…,n} ordered by inclusion, and a chain is a family of pairwise comparable elements of 2[n]. Let s=2n/0n⌊nfalse/2⌋, which is the average size of a chain in a minimal chain decomposition of 2[n]. We prove that 2[n] can be partitioned into 0n⌊nfalse/2⌋ chains such that all but at most o(1) proportion of the chains have size s(1+o(1)). This asymptotically proves a conjecture of Füredi from 1985. Our proof is based on probabilistic arguments. To analyze our random partition we develop a weighted variant of the graph container method. Using this result, we also answer a Kalai‐type question raised recently by Das, Lamaison, and Tran. What is the minimum number of forbidden comparable pairs forcing that the largest subfamily of 2[n] not containing any of them has size at most 0n⌊nfalse/2⌋? We show that the answer is (π8+o(1))2nn. Finally, we discuss how these uniform chain decompositions can be used to optimize and simplify various results in extremal set theory.

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Uniform chain decompositions and applications

Sudakov¹,

Tomon²,

Wagner³

2019

Preprint

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The Boolean lattice 2 [n] is the family of all subsets of [n] = {1, . . . , n} ordered by inclusion, and a chain is a family of pairwise comparable elements of 2 [n] . Let s = 2 n / n ⌊n/2⌋ , which is the average size of a chain in a minimal chain decomposition of 2 [n] . We prove that 2 [n] can be partitioned into n ⌊n/2⌋ chains such that all but at most o(1) proportion of the chains have size s(1 + o( 1)). This asymptotically proves a conjecture of Füredi from 1985. Our proof is based on probabilistic arguments. To analyze our random partition we develop a weighted variant of the graph container method.Using this result, we also answer a Kalai-type question raised recently by Das, Lamaison and Tran. What is the minimum number of forbidden comparable pairs forcing that the largest subfamily of 2 [n] not containing any of them has size at most n ⌊n/2⌋ ? We show that the answer is ( π 8 + o(1))2 n √ n. Finally, we discuss how these uniform chain decompositions can be used to optimize and simplify various results in extremal set theory.

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On Orthogonal Symmetric Chain Decompositions

Cited by 3 publications

References 38 publications

On 1-Factorizations of Bipartite Kneser Graphs

On 1-Factorizations of Bipartite Kneser Graphs

Uniform chain decompositions and applications

Uniform chain decompositions and applications

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