Euler paths and ends in automatic and recursive graphs

It is shown that the existence of an Euler path in a recursive graph is complete for the class DΣ 3 of all set differences of two Σ 3 sets. The same problem for highly recursive graphs as well as automatic graphs is shown to be Π 2 -complete. Moreover, the arithmetic level for bounding the number of ends in an automatic/recursive graph as well as computing the number of infinite paths in an automatic/recursive finitely branching tree is determined.