On extremal factors of de Bruijn-like graphs

TL;DR

This paper generalizes a classic result about maximum vertex-disjoint cycles in de Bruijn graphs to their tensor products with cycles, providing new counting formulas for various cycling register rules.

Contribution

It extends Mykkeltveit's theorem to tensor products of de Bruijn graphs and cycles, and develops counting formulas for cycling register rules.

Findings

Maximum vertex-disjoint cycles in tensor products of de Bruijn graphs are characterized.

Counting formulas for cycling register rules are derived.

The results include linear register rules proposed by Golomb.

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.