The maximum independent sets of de Bruijn graphs of diameter 3

TL;DR

This paper characterizes the maximum independent sets of de Bruijn graphs of diameter 3 for any alphabet size, providing a recurrence relation and generating function, and enabling the construction of large comma-free codes.

Contribution

It offers an inductive characterization and recurrence relation for maximum independent sets in de Bruijn graphs of diameter 3, extending understanding to arbitrary alphabet sizes.

Findings

Derived a recurrence relation for maximum independent sets.

Constructed exponentially many comma-free codes of length 3.

Provided an exponential generating function for their count.

Abstract

The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabet of size d, with edges between words which are distinct substrings of a word of length 4. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs B(d,3) and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.