Contributions to the theory of de Bruijn cycles

TL;DR

This paper extends the theory of de Bruijn cycles by demonstrating their existence for various combinatorial objects, including subsets and words with specific weights, broadening their applicability in combinatorics.

Contribution

It generalizes existing results by proving de Bruijn cycles can be constructed for words with weights in a specified interval, beyond previous cases.

Findings

De Bruijn cycles exist for words with weights between s and t.

Extension of de Bruijn cycles to subsets of [n] with size in [s,t].

Application of de Bruijn's theorem to labeled subposets of the Boolean lattice.

Abstract

A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so that each object appears exactly once as a set of consecutive elements in the cycle. In this paper, we show the power of de Bruijn's original theorem, namely that the cycles bearing his name exist for n-letter words on a k-letter alphabet for all values of k,n, to prove that we can create de Bruijn cycles for the assignment of elements of [n]={1,2,....,n} to the sets in any labeled subposet of the Boolean lattice; de Bruijn's theorem corresponds to the case when the subposet in question consists of a single ground element. The landmark work of Chung, Diaconis, and Graham extended the agenda of finding de Bruijn cycles to possibly the next most natural set of combinatorial objects, namely k-subsets of [n]. In this area, important contributions have been those of Hurlbert and Rudoy. Here we follow the direction of Blanca and Godbole, who proved that, in a suitable encoding, de Bruijn cycles can be created for the subsets of [n$ of size in the interval [s,t]; 0<=s<t<=n$. In this paper we generalize this result to exhibit existence of de Bruijn cycles for words with weight between s and t, where these parameters are suitably restricted.