We investigate the complexity of counting Eulerian tours (#ET) and its variations from two perspectives–the complexity of exact counting and the complexity w.r.t. approximation-preserving reductions (Dyer et al., 2004). We prove that #ET is #P-complete even for planar 4-regular graphs. A closely related problem is that of counting A-trails (#A-trails) in graphs with rotational embedding schemes (so called maps). Kotzig (1968) showed that #A-trails can be computed in polynomial time for 4-regular plane graphs (embedding in the plane is equivalent to giving a rotational embedding scheme). We show that for 4-regular maps the problem is #P-hard. Moreover, we show that from the approximation viewpoint #A-trails in 4-regular maps captures the essence of #ET, that is, we give an AP-reduction from #ET in general graphs to #A-trails in 4-regular maps. The reduction uses a fast mixing result for a card shuffling problem (Wilson, 2004).
