Eulerian Digraphs and Oriented Trees

Abstract

A famous problem which goes back to Euler asks for what graphs G is there a closed walk which uses every edge exactly once. (There is also a version for non-closed walks.) Such a walk is called an Eulerian tour (also known as an Eulerian cycle). A graph which has an Eulerian tour is called an Eulerian graph. Euler’s famous theorem (the first real theorem of graph theory) states that a graph G without isolated vertices (which clearly would be irrelevant) is Eulerian if and only if it is connected and every vertex has even degree. Here we will be concerned with the analogous theorem for directed graphs. We want to know not just whether an Eulerian tour exists, but also how many there are. We will prove an elegant determinantal formula for this number closely related to the Matrix-Tree Theorem. For the case of undirected graphs no analogous formula is known, explaining why we consider only the directed case.