Eulerian Directed Multigraphs

TL;DR

This paper characterizes the existence of Eulerian circuits and paths in finite connected directed multigraphs using conditions on vertex degrees and vertex pair connectivity.

Contribution

It provides necessary and sufficient conditions for directed multigraphs to have Eulerian circuits and paths, extending classical graph theory results.

Findings

Characterization of Eulerian circuits in directed multigraphs.

Conditions for existence of Eulerian paths in directed multigraphs.

Theoretical criteria involving vertex degrees and connectivity.

Abstract

For $\Delta$ a finite connected nontrivial directed multigraph, we prove: 1. $\Delta$ has a directed circuit using each directed edge exactly once if and only if both each pair of distinct vertices of $\Delta$ occur in a common directed circuit and in-degree$({\bf x}) =$ out-degree$({\bf x})$ for every vertex ${\bf x}$. 2. $\Delta$ contains a non-circuit directed path which uses every directed edge exactly once if and only if both every pair of distinct vertices of $\Delta$ occur in a common directed circuit and there are vertices ${\bf b \not= e}$ such that in-degree$({\bf e}) -$ out-degree$({\bf e}) = 1 =$ out-degree$({\bf b}) -$ in-degree$({\bf b})$ but, for every vertex ${\bf x \notin \{b,e\}}$, it happens that in-degree$({\bf x}) =$ out-degree$({\bf x})$.