On the central levels problem

@article{Gregor2020OnTC,
  title={On the central levels problem},
  author={Petr Gregor and Ondrej Micka and Torsten M{\"u}tze},
  journal={ArXiv},
  year={2020},
  volume={abs/1912.01566}
}
The central levels problem asserts that the subgraph of the $(2m+1)$-dimensional hypercube induced by all bitstrings with at least $m+1-\ell$ many 1s and at most $m+\ell$ many 1s, i.e., the vertices in the middle $2\ell$ levels, has a Hamilton cycle for any $m\geq 1$ and $1\le \ell\le m+1$. This problem was raised independently by Savage, by Gregor and Skrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case $\ell=1$, and… 

Gray codes and symmetric chains

On a combinatorial generation problem of Knuth

This work proves Knuth's conjecture in a more general form, allowing for arbitrary shifts $s\geq 1$ that are coprime to $2n+1$, and presents an algorithm to compute this ordering, generating each new bitstring in $\mathcal{O}(n)$ time.

Combinatorial Gray codes-an updated survey

This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes and gives an update on Savage’s influential survey, incorporating many more recent developments.

References

SHOWING 1-10 OF 70 REFERENCES

A Constant-Time Algorithm for Middle Levels Gray Codes

This work presents an algorithm for computing a middle levels Gray code in optimal time and space: each new set in the Gray code is generated in time $${{\mathcal {O}}}(1)$$ O ( 1 ) on average, and the required space is  $${{\ mathcal { O}}}(n)$ O ( n ) .

Monotone Gray Codes and the Middle Levels Problem

Hamiltonian Cycles and Symmetric Chains in Boolean Lattices

Let B(n) be the subset lattice of {1,2,⋯,n}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}

Trimming and gluing Gray codes

A short proof of the middle levels theorem

A new proof of the well-known middle levels conjecture is presented, which is much shorter and more accessible than the original proof.

On generalized middle-level problem

Hamilton Cycles that Extend Transposition Matchings in Cayley Graphs of Sn

It is shown that, for any basis B of transpositions for S_n, there is a Hamilton cycle in Cay( B:S_n )$ that includes every edge of $M_b $ in Cay, and that it is possible to generate all permutations of $1,2, \ldots ,n$ byTranspositions in B so that every other transposition is b.

On a conjecture of Füredi

Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture

It is proved that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.
...