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To Infinity and Beyond: Continuing De Bruijn Sequences by Extending the Alphabet

Yotam Svoray, Gera Weiss

TL;DR

This paper tackles the problem of extending a De Bruijn sequence of order $n$ over the alphabet $[k]$ to an order-$n$ sequence over the expanded alphabet $[k+1]$, and shows how to create an infinite De Bruijn sequence by iteratively increasing the alphabet. The central approach introduces the $(k,n)$-Prefer Max De Bruijn sequence with a recursive rule and proves the Onion Theorem, which states that the $(k-1,n)$-prefer-max sequence is a suffix of the $(k,n)$-prefer-max sequence, enabling an infinite construction over the nonnegative integers. It further defines Onion De Bruijn sequences via Layered De Bruijn graphs, analyzes their relation to the Prefer Max family, and proves a structure property that such sequences align with the prefer-max prefixes for words of length $n-1$. Overall, the results provide a constructive path to infinite De Bruijn sequences and establish a taxonomy of onion sequences, including non-uniqueness phenomena.

Abstract

This article presents proof that the reverse of the Prefer Max De Bruijn sequence can be expanded into an infinite De Bruijn sequence by increasing the size of the alphabet. Furthermore, we show that every De Bruijn sequence possessing this characteristic exhibits behavior similar to that of the reverse of the Prefer Max De Bruijn sequence.

To Infinity and Beyond: Continuing De Bruijn Sequences by Extending the Alphabet

TL;DR

This paper tackles the problem of extending a De Bruijn sequence of order over the alphabet to an order- sequence over the expanded alphabet , and shows how to create an infinite De Bruijn sequence by iteratively increasing the alphabet. The central approach introduces the -Prefer Max De Bruijn sequence with a recursive rule and proves the Onion Theorem, which states that the -prefer-max sequence is a suffix of the -prefer-max sequence, enabling an infinite construction over the nonnegative integers. It further defines Onion De Bruijn sequences via Layered De Bruijn graphs, analyzes their relation to the Prefer Max family, and proves a structure property that such sequences align with the prefer-max prefixes for words of length . Overall, the results provide a constructive path to infinite De Bruijn sequences and establish a taxonomy of onion sequences, including non-uniqueness phenomena.

Abstract

This article presents proof that the reverse of the Prefer Max De Bruijn sequence can be expanded into an infinite De Bruijn sequence by increasing the size of the alphabet. Furthermore, we show that every De Bruijn sequence possessing this characteristic exhibits behavior similar to that of the reverse of the Prefer Max De Bruijn sequence.

Paper Structure

This paper contains 4 sections, 3 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Notations
  3. An infinite De Bruijn sequence
  4. Onion De Bruijn sequences

Key Result

Theorem 1

Every De Bruijn order sequence $n$ in at least three symbols can be extended to a De Bruijn sequence of order $n + 1$. Every De Bruijn sequence of order $n$ in two symbols can not be extended to order $n + 1$, but it can be extended to order $n + 2$.

Theorems & Definitions (8)

  • Theorem : Theorem $1$ in becher2011extending
  • definition 1
  • Theorem 1: Onion Theorem
  • proof
  • definition 2
  • Theorem 2
  • proof
  • remark 1