We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a GapL 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 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.