The Prism Over the Middle-levels Graph is Hamiltonian

Horák, Peter; Kaiser, Tomáš; Rosenfeld, Moshe; Ryjáček, Zdeněk

doi:10.1007/s11083-005-9008-7

Cited by 20 publications

(14 citation statements)

References 5 publications

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“…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. On the other hand, it has been shown that M 2k+1 has a closed spanning walk in which no edge and no vertex occurs more than twice [5].…”

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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Section: Introductionmentioning

confidence: 99%

An update on the middle levels problem

Shields

Savage

2009

Discrete Mathematics

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“…In [15] we proved that the middle-levels graph is prism-hamiltonian. Nash-Williams [20] conjectured that 4-connected, 4-regular graphs are hamiltonian.…”

Section: Problem 1 Let G Be a Graph Of Order N And Let X And Y Be Twmentioning

confidence: 92%

Hamilton cycles in prisms

Kaiser

Ryjáček

Kráľ

et al. 2007

Journal of Graph Theory

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“…Savage and Winkler [15] showed that B k has a cycle containing at least 86.7% of the graph vertices, for k ≥ 19. Horák, Kaiser, Rosenfeld and Ryjáček [7] showed the middle levels graph has a closed spanning 2-trail. It remains open to prove similar results for the odd graphs.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…Horák, Kaiser, Rosenfeld and Ryjáček [7] showed that the prism over the middle levels graph is Hamiltonian, and consequently B k has a closed spanning 2-trail. 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.…”

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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The Prism Over the Middle-levels Graph is Hamiltonian

Abstract: Let B k be the bipartite graph defined by the subsets of {1, . . . , 2k + 1} of size k and k + 1. We prove that the prism over B k is hamiltonian. We also show that B k has a closed spanning 2-trail.

Cited by 20 publications

References 5 publications

An update on the middle levels problem

An update on the middle levels problem

Hamilton cycles in prisms

Hamiltonian paths in odd graphs

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