In graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in directed (oriented) graphs. The name is an acronym of the names of people who discovered it: de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte.
Let G = (V, E) be a directed graph. An Eulerian circuit is a directed closed path which visits each edge exactly once. In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian. We denote the indegree of a vertex v by deg(v).
The BEST theorem states that the number ec(G) of Eulerian circuits in a connected Eulerian graph G is given by the formula
Here tw(G) is the number of arborescences, which are trees directed towards the root at a fixed vertex w in G. The number tw(G) can be computed as a determinant, by the version of the matrix tree theorem for directed graphs. It is a property of Eulerian graphs that tv(G) = tw(G) for every two vertices v and w in a connected Eulerian graph G.
The BEST theorem shows that the number of Eulerian circuits in directed graphs can be computed in polynomial time, a problem which is #P-complete for undirected graphs. It is also used in the asymptotic enumeration of Eulerian circuits of complete and complete bipartite graphs.
The BEST theorem was first stated in this form in a "note added in proof" to the paper of van Aardenne-Ehrenfest and de Bruijn (1951). The original proof was bijective and generalized the de Bruijn sequences. It is a variation on an earlier result by Smith and Tutte (1941).
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