Gray codes and symmetric chains

TL;DR

This paper explores cyclic listings of specific bitstrings within hypercubes, generalizing the middle two levels problem, and introduces new constructions of symmetric chain decompositions with applications to combinatorial structures.

Contribution

It provides solutions for constructing cyclic listings for certain hypercube levels and introduces new methods for symmetric chain decompositions, including edge-disjoint decompositions.

Findings

Constructed a cyclic listing for the case ll=2.

Developed cycle factors for general ll values.

Created four edge-disjoint symmetric chain decompositions for hypercubes.

Abstract

We consider the problem of constructing a cyclic listing of all bitstrings of length $2n+1$ with Hamming weights in the interval $[n+1-\ell,n+\ell]$, where $1\leq \ell\leq n+1$, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (the case $\ell=1$). We provide a solution for the case $\ell=2$, and we solve a relaxed version of the problem for general values of $\ell$, by constructing cycle factors for those instances. The proof of the first result uses the lexical matchings introduced by Kierstead and Trotter, which we generalize to arbitrary consecutive levels of the hypercube. The proof of the second result uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets. We also present several new constructions of such decompositions based on lexical matchings. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the $n$-dimensional hypercube for any $n\geq 12$.