A short proof of the middle levels theorem

Gregor, Petr; Mütze, Torsten; Nummenpalo, Jerri

doi:10.19086/da.3659

Cited by 25 publications

(63 citation statements)

References 10 publications

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“…This paper complements previous work [5] on reinterpreting the middle-levels theorem [6,8] via a numeral system that enumerates all ordered trees. Let 0 < k ∈ Z and let n = 2k + 1.…”

Section: Introductionsupporting

confidence: 60%

A numeral system for the middle-levels graphs

Dejter

2021

EJGTA

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“…This paper complements previous work [5] on reinterpreting the middle-levels theorem [6,8] via a numeral system that enumerates all ordered trees. Let 0 < k ∈ Z and let n = 2k + 1.…”

Section: Introductionsupporting

confidence: 60%

A numeral system for the middle-levels graphs

Dejter

2021

EJGTA

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“…By this definition, every pair of almostorthogonal chain decompositions is edge-disjoint, but not necessarily vice versa. The main application of edge-disjoint chain decompositions in [GJM + 18] was to construct cycle factors in subgraphs of Q n induced by an interval of levels around the middle, with the goal of generalizing the recent proof of the middle levels conjecture by Mütze [Müt16] (see also [GMN18]). It is also easy to check that Q n admits at most b n pairwise edge-disjoint chain decompositions.…”

Section: Our Resultsmentioning

confidence: 99%

On Orthogonal Symmetric Chain Decompositions

Däubel

Jäger

Mütze³

et al. 2019

Electron. J. Combin.

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The n-cube is the poset obtained by ordering all subsets of {1, . . . , n} by inclusion, and it can be partitioned into n n/2 chains, which is the minimum possible number. Two such decompositions of the n-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the ncube has n/2 + 1 pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the n-cube has three pairwise orthogonal chain decompositions for n ≥ 24. In this paper, we construct four pairwise orthogonal chain decompositions of the n-cube for n ≥ 60. We also construct five pairwise edge-disjoint symmetric chain decompositions of the n-cube for n ≥ 90, where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result

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“…It was hoped that people can find two 1-factors which form a Hamiltonian cycle [17]. Yet after extensive studies for thirty years the conjecture itself was settled by Mütze [21]; see also [14] for a recent and shorter proof and see [22] for an optimal algorithm for computing such a Hamiltonian cycle. Moreover, Mütze and Su [23] settles the Hamiltonian problem for all the bipartite Kneser graphs.…”

Section: Motivation and Related Workmentioning

confidence: 99%

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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A short proof of the middle levels theorem

Cited by 25 publications

References 10 publications

A numeral system for the middle-levels graphs

A numeral system for the middle-levels graphs

On Orthogonal Symmetric Chain Decompositions

On 1-Factorizations of Bipartite Kneser Graphs

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