De Bruijn graph

In graph theory, an n-dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols. It has mⁿvertices, consisting of all possible length-n sequences of the given symbols; the same symbol may appear multiple times in a sequence. If we have the set of m symbols ${\displaystyle S:=\{s_{1},\dots ,s_{m}\}}$ then the set of vertices is:

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 (aka directed edges) is

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.

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 ${\displaystyle n}$-dimensional De Bruijn graph corresponds to an edge of the ${\displaystyle (n-1)}$-dimensional De Bruijn graph, and each edge in the ${\displaystyle n}$-dimensional De Bruijn graph corresponds to a two-edge path in the ${\displaystyle (n-1)}$-dimensional De Bruijn graph.

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