BEST theorem

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 these in- and out-degree 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 t_w(G) is the number of arborescences, which are trees directed towards the root at a fixed vertex w in G. The number t_w(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 t_v(G) = t_w(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.

...
Wikipedia