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On extremal factors of de Bruijn-like graphs

Nicolás Álvarez, Verónica Becher, Martín Mereb, Ivo Pajor, Carlos Miguel Soto

TL;DR

The paper generalizes Golomb's extremal cycle partition result from de Bruijn graphs to astute graphs $G_{n,k}$, the tensor product of a de Bruijn graph with a cycle, and develops a comprehensive counting framework for cycling-register–generated factors. It proves extremality of the pure cycling register factor $F_k(r_n)$ when $k|n$ or $n|k$, and provides explicit counting formulas for a broad class of affine succession rules, including linear Golomb rules, via Burnside’s lemma and polynomial gcd analysis. It further derives concrete corollaries for the PCR, incremented PCR, and Xor cycling rules, giving closed expressions in terms of $n$, $k$, the alphabet size $b$, and gcd-related parameters. The method combines discrete Fourier analysis, finite-field/ring polynomial gcd, and affine necklace enumeration to yield a unified combinatorial approach to cycle decompositions in de Bruijn–type graphs with practical implications for sequence design and graph factorization.

Abstract

In 1972 Mykkeltveit proved that the maximum number of vertex-disjoint cycles in the de Bruijn graphs of order $n$ is attained by the pure cycling register rule, as conjectured by Golomb. We generalize this result to the tensor product of the de Bruijn graph of order $n$ and a simple cycle of size $k$, when $n$ divides $k$ or vice versa. We also develop counting formulae for a large family of cycling register rules, including the linear register rules proposed by Golomb.

On extremal factors of de Bruijn-like graphs

TL;DR

The paper generalizes Golomb's extremal cycle partition result from de Bruijn graphs to astute graphs , the tensor product of a de Bruijn graph with a cycle, and develops a comprehensive counting framework for cycling-register–generated factors. It proves extremality of the pure cycling register factor when or , and provides explicit counting formulas for a broad class of affine succession rules, including linear Golomb rules, via Burnside’s lemma and polynomial gcd analysis. It further derives concrete corollaries for the PCR, incremented PCR, and Xor cycling rules, giving closed expressions in terms of , , the alphabet size , and gcd-related parameters. The method combines discrete Fourier analysis, finite-field/ring polynomial gcd, and affine necklace enumeration to yield a unified combinatorial approach to cycle decompositions in de Bruijn–type graphs with practical implications for sequence design and graph factorization.

Abstract

In 1972 Mykkeltveit proved that the maximum number of vertex-disjoint cycles in the de Bruijn graphs of order is attained by the pure cycling register rule, as conjectured by Golomb. We generalize this result to the tensor product of the de Bruijn graph of order and a simple cycle of size , when divides or vice versa. We also develop counting formulae for a large family of cycling register rules, including the linear register rules proposed by Golomb.
Paper Structure (14 sections, 14 theorems, 80 equations, 2 figures)

This paper contains 14 sections, 14 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction and statement of results
  2. Proof of Theorem \ref{['thm:extremal']}
  3. First case: There exists some string $s_i$ such that $C(s_i)$ is not real.
  4. Second case: The transform $C(s_i)$ is $0$ for all $i$.
  5. Third case: The transform $C(s_i)$ is always real, but not always $0$.
  6. Proof of Theorem \ref{['thm:count']}
  7. Burnside's Lemma
  8. GCDs of certain families of polynomials
  9. Affine Succession Rules
  10. Burnside's Lemma for affine necklaces
  11. Proof of the Corollaries
  12. Proof of Corollary \ref{['cor:coro1']}
  13. Proof of Corollary \ref{['thm:coro2']}
  14. Proof of Corollary \ref{['cor:coro3']}

Key Result

Theorem 1

Let $n$ and $k$ be positive integers such that $k$ divides $n$ or $n$ divides $k$. The factor $F_{k}(r_n)$ produced by the pure cycling register rule $r_n$ is extremal.

Figures (2)

  • Figure 1: Pure Cycling Register induced factors may not be extremal in astute graphs
  • Figure 2: Transforms of the strings on the PCR cycle generated by $s_0=123351$. The large red point is the distinguished vertex for this PCR cycle.

Theorems & Definitions (40)

  • Definition : Astute graph
  • Remark 1
  • Definition : Factor
  • Definition : Succession rule
  • Remark 2
  • Definition : Action of a succession rule on astute graphs
  • Definition : Factor generated by succession rule
  • Definition : Affine relation
  • Definition : Affine succession rule
  • Remark 3
  • ...and 30 more