  We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
  Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
  While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize $\#SAC^1$) is relatively easy, establishing a uniform
$\#SAC^1$ bound needs a careful use of polynomial interpolation.
