Corpus ID: 15975960

Long Cycles in the Middle Two Levels of the Boolean Lattice

@article{Savage1997LongCI,
  title={Long Cycles in the Middle Two Levels of the Boolean Lattice},
  author={Carla D. Savage},
  journal={Ars Combinatoria},
  year={1997}
}
  • C. Savage
  • Published 1997
  • Mathematics
  • Ars Combinatoria
An intriguing open question is whether the graph formed by the middle two levels of the Boolean lattice of subsets of a k element set has a Hamilton path for all k We consider nding a lower bound on the length of the longest cycle in this graph A result of Babai for vertex transitive graphs gives a lower bound of N where N is the total number of vertices in the middle two levels In this paper we show how to construct a cycle of length N c where c 
A short proof of the middle levels theorem
TLDR
A new proof of the well-known middle levels conjecture is presented, which is much shorter and more accessible than the original proof. Expand
On Hamilton cycles in highly symmetric graphs
TLDR
Some of the recent results on Hamilton cycles in various families of highly symmetric graphs are surveyed, including the solution of the well-known middle levels conjecture, and several far-ranging generalizations of it that were proved subsequently. Expand
Proof of the middle levels conjecture
Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length 2n + 1 that have exactly n or n + 1 entries equal to 1, with an edge between any two vertices forExpand
Proof of the middle levels conjecture
Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices forExpand
Hamilton cycle heuristics in hard graphs
In this thesis, we use computer methods to investigate Hamilton cycles and paths in several families of graphs where general results are incomplete, including Kneser graphs, cubic Cayley graphs andExpand
A Hamilton Path Heuristic with Applicationsto the Middle Two Levels
The notorious middle two levels problem is to nd a Hamilton cycle in the middle two levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially ordered set of subsets of a 2k + 1-element setExpand
A Hamilton Path Heuristic with Applications to the Middle Two Levels Problem
The notorious middle two levels problem is to nd a Hamilton cycle in the middle two levels, M2k+1, of the Hasse diagram ofB2k+1 (the partially ordered set of subsets of a 2k + 1-element set orderedExpand
Monotone Gray Codes and the Middle Levels Problem
TLDR
It is shown that for every ϵ > 0, there is an h ⩾ 1 so that if a hamilton cycle exists in the middle two levels of B 2k + 1 for 1 ⩽ k⩽ h, then there is a cycle of length at least (1 − ϵ) N(k) for all k ⩵ 1, where N( k)=2(2kk+1). Expand
Bipartite Kneser graphs are Hamiltonian
TLDR
It is established the existence of long cycles in Kneser graphs (visiting almost all vertices), generalizing and improving upon previous results on this problem. Expand
An update on the middle levels problem
TLDR
The result was achieved by an algorithmic improvement that made it possible to find a Hamilton path in a reduced graph having 129,644,790 vertices, using a 64-bit personal computer. Expand
...
1
2
3
...

References

SHOWING 1-10 OF 11 REFERENCES
Explicit matchings in the middle levels of the Boolean lattice
New classes of explicit matchings for the bipartite graph ℬ(k) consisting of the middle two levels of the Boolean lattice on 2k+1 elements are constructed and counted. This research is part of anExpand
Lexicographic matchings cannot form Hamiltonian cycles
For any positive integer k let B(k) denote the bipartite graph of k- and k+1-element subsets of a 2k+1-element set with adjacency given by containment. It has been conjectured that for all k, B(k) isExpand
Long cycles in vertex-transitive graphs
  • L. Babai
  • Mathematics, Computer Science
  • J. Graph Theory
  • 1979
We prove that every connected vertex-transitive graph on n ≥ 4 vertices has a cycle longer than (3n)1/2. The correct order of magnitude of the longest cycle seems to be a very hard question.
Lexicographic matching in Boolean algebras
Abstract Hall's theorem on systems of distinct representatives implies the existence of matchings between two consecutive levels of a Boolean algebra. In this paper a very simple construction of suchExpand
The antipodal layers problem
  • G. Hurlbert
  • Computer Science, Mathematics
  • Discret. Math.
  • 1994
TLDR
Here it is shown that the conjecture holds for n bigger than roughly k2, with k large enough, and a new product between ranked posets is defined, giving rise to many new representations of M 2k+1,k. Expand
An Algorithm for Generating Subsets of Fixed Size With a Strong Minimal Change Property
Presentation d'un algorithme pour generer une liste S 1 , S 2 ,...,Sm (m=( k n )) de sous-ensembles de taille k d'un ensemble de taille n
\Long cycles in revolving door graphs
  • Congressus Numerantium
  • 1987
Reingold, \EEcient generation of the binary reeected Gray code and its applications
  • Communications of the ACM
  • 1976
Problem 11 in Combinatorial Structures and their Applications, G o r den and Breach
  • Problem 11 in Combinatorial Structures and their Applications, G o r den and Breach
  • 1970
...
1
2
...