A Cycle Joining Construction of the Prefer-Max De Bruijn Sequence

TL;DR

This paper introduces a new cycle joining method for constructing the prefer-max De Bruijn sequence, simplifying proofs of key properties and related theorems in the field.

Contribution

It presents a novel cycle joining construction that offers a straightforward proof of the prefer-max De Bruijn sequence's properties and related theorems.

Findings

Validates the onion theorem for prefer-max sequences

Confirms the correctness of shift rules for prefer-max and prefer-min

Provides an alternative proof for the FKM-theorem

Abstract

We propose a novel construction for the well-known prefer-max De Bruijn sequence, based on the cycle joining technique. We further show that the construction implies known results from the literature in a straightforward manner. First, it implies the correctness of the onion theorem, stating that, effectively, the reverse of prefer-max is in fact an infinite De Bruijn sequence. Second, it implies the correctness of recently discovered shift rules for prefer-max, prefer-min, and their reversals. Lastly, it forms an alternative proof for the seminal FKM-theorem.