On the central levels problem

  title={On the central levels problem},
  author={Petr Gregor and Ondrej Micka and Torsten M{\"u}tze},
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.



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

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

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 for which

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

  • I. Tomon
  • Mathematics, Computer Science
    Eur. J. Comb.
  • 2015

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.

A Survey of Combinatorial Gray Codes

The area of combinatorial Gray codes is surveyed, recent results, variations, and trends are described, and some open problems are highlighted.