  In this paper we obtain the expectation and variance of the number of Euler
tours of a random Eulerian directed graph with fixed out-degree sequence. We
use this to obtain the asymptotic distribution of the number of Euler tours of
a random $d$-in/$d$-out graph and prove a concentration result. We are then
able to show that a very simple approach for uniform sampling or approximately
counting Euler tours yields algorithms running in expected polynomial time for
almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de
Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of
Euler tours of an Eulerian directed graph with out-degree sequence $\mathbf{d}$
is the product of the number of arborescences and the term
$\frac{1}{n}[\prod_{v \in V}(d_v-1)!]$. Therefore most of our effort is towards
estimating the moments of the number of arborescences of a random graph with
fixed out-degree sequence.

Discrete Mathematics;Data Structures and Algorithms;Combinatorics;Probability