# 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.

## Precise statement

Let G = (VE) 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

${\displaystyle \operatorname {ec} (G)=t_{w}(G)\prod _{v\in V}{\bigl (}\deg(v)-1{\bigr )}!.}$

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.

## Applications

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.[1] It is also used in the asymptotic enumeration of Eulerian circuits of complete and complete bipartite graphs.[2][3]

## History

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).

## Notes

1. ^ Brightwell and Winkler, "Note on Counting Eulerian Circuits", CDAM Research Report LSE-CDAM-2004-12, 2004.
2. ^ Brendan McKay and Robert W. Robinson, Asymptotic enumeration of eulerian circuits in the complete graph, Combinatorica, 10 (1995), no. 4, 367–377.
3. ^ M.I. Isaev, Asymptotic number of Eulerian circuits in complete bipartite graphs Archived 2010-04-15 at the Wayback Machine (in Russian), Proc. 52-nd MFTI Conference (2009), Moscow.

## References

• Euler, L. (1736), "Solutio problematis ad geometriam situs pertinentis", Commentarii Academiae Scientiarum Petropolitanae (in Latin), 8: 128–140.
• Tutte, W. T.; Smith, C. A. B. (1941), "On unicursal paths in a network of degree 4", American Mathematical Monthly, 48: 233–237, doi:10.2307/2302716, JSTOR 2302716.
• van Aardenne-Ehrenfest, T.; de Bruijn, N. G. (1951), "Circuits and trees in oriented linear graphs", Simon Stevin, 28: 203–217.