Long cycles in the middle two layers of the discrete cube

Johnson, J. Robert

doi:10.1016/j.jcta.2003.11.004

Cited by 34 publications

(27 citation statements)

References 3 publications

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“…, k}. This approximate version of the conjecture is similar in spirit to the line of work [Sav93,FT95,SW95,Joh04] that preceded the proof of Theorem 1. 1.2.…”

supporting

confidence: 60%

Trimming and gluing Gray codes

Gregor

Mütze

2018

Theoretical Computer Science

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Abstract. We consider the algorithmic problem of generating each subset of [n] := {1, 2, . . . , n} whose size is in some interval [k, l], 0 ≤ k ≤ l ≤ n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k = 0 and l = n this is the classical problem of generating all 2 n subsets of [n] by element additions/removals, and for k = l this is the classical problem of generating all n k subsets of [n] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n, k, and l, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant O(1) time per generated set and O(n) space. Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n-dimensional cube Q n induced by all levels [k, l]. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q 2k+1 induced by all levels [k − c, k + 1 + c], c ∈ {0, 1, . . . , k}, has a Hamilton cycle. We also prove an approximate version of this generalized conjecture, showing that this graph has a cycle that visits a (1 − o(1))-fraction of all vertices.

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“…, k}. This approximate version of the conjecture is similar in spirit to the line of work [Sav93,FT95,SW95,Joh04] that preceded the proof of Theorem 1. 1.2.…”

supporting

confidence: 60%

Trimming and gluing Gray codes

Gregor

Mütze

2018

Theoretical Computer Science

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“…Recently, it was shown that M 2k+1 is "asymptotically Hamiltonian" in the following sense [7]: There is a constant c such that for all k, M 2k+1 has a cycle of length at least (1 − c/ √ k)V (M 2k+1 ). The existence proof in [7] does not attempt to estimate c. To improve the bound in Theorem 1 for, say all k < 5600, would require a value of c < 10.…”

Section: Introductionmentioning

confidence: 99%

An update on the middle levels problem

Shields

Savage

2009

Discrete Mathematics

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The middle levels problem is to find a Hamilton cycle in the middle levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially-ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known, from [I. Shields, C.D. Savage, A Hamilton path heuristic with applications to the middle two levels problem, in: ], was that M 2k+1 is Hamiltonian for all positive k through k = 15. In this note we announce that M 33 and M 35 have Hamilton cycles. The result was achieved by an algorithmic improvement that made it possible to find a Hamilton path in a reduced graph (of complementary necklace pairs) having 129,644,790 vertices, using a 64-bit personal computer.

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“…Savage and Winkler [15] showed that if B k has a Hamiltonian cycle for k ≤ h, then B k has a cycle containing a fraction 1 − ε of the graph vertices for all k, where ε is a function of h. For example, since B k has a Hamiltonian cycle for 2 ≤ k ≤ 18, then B k has a cycle containing at least 86.7% of the graph vertices, for any k ≥ 2. Johnson [8] showed that both B k and O k have cycles containing (1 − o(1))|V | vertices, where |V | is the number of vertices in the graph. Johnson and Kierstead [9] used matchings of B k to show that O k has not only a 2-factor (spanning collection of cycles), but also a 2-factorisation.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian paths in odd graphs

Bueno

Faria

Figueiredo

et al. 2009

APPL ANAL DISCRETE M

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3Lovász conjectured that every connected vertex-transitive graph has a Hamiltonian path. The odd graphs O k form a well-studied family of connected, k-regular, vertex-transitive graphs. It was previously known that O k has Hamiltonian paths for k ≤ 14. A direct computation of Hamiltonian paths in O k is not feasible for large values of k, because O k has 2k − 1 k − 1 vertices and k 2 2k − 1 k − 1 edges. We show that O k has Hamiltonian paths for 15 ≤ k ≤ 18. Instead of directly running any heuristics, we use existing results on the middle levels problem, therefore further relating these two fundamental problems, namely finding a Hamiltonian path in the odd graph and finding a Hamiltonian cycle in the corresponding middle levels graph. We show that further improved results for the middle levels problem can be used to find Hamiltonian paths in O k for larger values of k.

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Long cycles in the middle two layers of the discrete cube

Cited by 34 publications

References 3 publications

Trimming and gluing Gray codes

Trimming and gluing Gray codes

An update on the middle levels problem

Hamiltonian paths in odd graphs

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