Eulerian Directed Multigraphs

Abstract

For $\hat{I}”$ a finite connected nontrivial directed multigraph, we prove: 1. $\hat{I}”$ has a directed circuit using each directed edge exactly once if and only if both each pair of distinct vertices of $\hat{I}”$ occur in a common directed circuit and in-degree$(\mathbf{x})=$ out-degree$(\mathbf{x})$ for every vertex $\mathbf{x}$. 2. $\hat{I}”$ contains a non-circuit directed path which uses every directed edge exactly once if and only if both every pair of distinct vertices of $\hat{I}”$ occur in a common directed circuit and there are vertices $\mathbf{b}\ne \mathbf{e}$ such that in-degree$(\mathbf{e})-$ out-degree$(\mathbf{e})=1=$ out-degree$(\mathbf{b})-$ in-degree$(\mathbf{b})$ but, for every vertex $\mathbf{x}\notin \{\mathbf{b},\mathbf{e}\}$, it happens that in-degree$(\mathbf{x})=$ out-degree$(\mathbf{x})$.