DeepAI
Log In Sign Up

On the central levels problem

12/03/2019
by   Petr Gregor, et al.
0

The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-ℓ many 1s and at most m+ℓ many 1s, i.e., the vertices in the middle 2ℓ levels, has a Hamilton cycle for any m≥ 1 and 1<ℓ< m+1. This problem was raised independently by Savage, by Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case ℓ=1, and classical binary Gray codes, namely the case ℓ=m+1. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of ℓ consecutive levels in the n-dimensional hypercube for any n> 1 and 1<ℓ< n+1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n-dimensional hypercube, n≥ 2, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code.

READ FULL TEXT

page 4

page 24

02/16/2018

Gray codes and symmetric chains

We consider the problem of constructing a cyclic listing of all bitstrin...
07/28/2019

Avoidable Vertices and Edges in Graphs

A vertex in a graph is simplicial if its neighborhood forms a clique. We...
10/01/2020

On the recursive structure of multigrid cycles

A new non-adaptive recursive scheme for multigrid algorithms is introduc...
08/17/2020

Optimal minimal Linear codes from posets

Recently, some infinite families of minimal and optimal binary linear co...
04/04/2018

A Euclidean Algorithm for Binary Cycles with Minimal Variance

The problem is considered of arranging symbols around a cycle, in such a...
08/17/2021

Star transposition Gray codes for multiset permutations

Given integers k≥ 2 and a_1,…,a_k≥ 1, let a:=(a_1,…,a_k) and n:=a_1+⋯+a_...
03/07/2020

Classification of minimally unsatisfiable 2-CNFs

We consider minimally unsatisfiable 2-CNFs, i.e., minimally unsatisfiabl...