De Bruijn graph

In graph theory, an formula_1-dimensional De Bruijn graph of formula_2 symbols is a directed graph representing overlaps between sequences of symbols. It has formula_3 vertices, consisting of all possible length-formula_1 sequences of the given symbols; the same symbol may appear multiple times in a sequence. For a set of formula_2 symbols formula_6, the set of vertices is:
formula_7
If one of the vertices can be expressed as another vertex by shifting all its symbols by one place to the left and adding a new symbol at the end of this vertex, then the latter has a directed edge to the former vertex. Thus the set of arcs (that is, directed edges) is
formula_8
Although De Bruijn graphs are named after Nicolaas Govert de Bruijn, they were discovered independently by both De Bruijn and I. J. Good. Much earlier, Camille Flye Sainte-Marie implicitly used their properties.
Properties.
The line graph construction of the three smallest binary De Bruijn graphs is depicted below. As can be seen in the illustration, each vertex of the formula_1-dimensional De Bruijn graph corresponds to an edge of the formula_14-dimensional De Bruijn graph, and each edge in the formula_1-dimensional De Bruijn graph corresponds to a two-edge path in the formula_14-dimensional De Bruijn graph.
Dynamical systems.
Binary De Bruijn graphs can be drawn in such a way that they resemble objects from the theory of dynamical systems, such as the Lorenz attractor:
This analogy can be made rigorous: the formula_1-dimensional formula_2-symbol De Bruijn graph is a model of the Bernoulli map
formula_21
The Bernoulli map (also called the 2x mod 1 map for formula_22) is an ergodic dynamical system, which can be understood to be a single shift of a -adic number. The trajectories of this dynamical system correspond to walks in the De Bruijn graph, where the correspondence is given by mapping each real formula_23 in the interval formula_24 to the vertex corresponding to the first formula_1 digits in the base-formula_2 representation of formula_23. Equivalently, walks in the De Bruijn graph correspond to trajectories in a one-sided subshift of finite type.
Embeddings resembling this one can be used to show that the binary De Bruijn graphs have queue number 2 and that they have book thickness at most 5.