The Maximum Independent Sets of de Bruijn Graphs of Diameter 3

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.