Multi-Eulerian tours of directed graphs

TL;DR

This paper introduces multi-Eulerian tours in directed graphs, a generalization of Eulerian tours, and establishes bounds on their minimal length based on a new measure called the Pham index, extending classical graph theory results.

Contribution

It defines multi-Eulerian tours for strongly connected graphs and generalizes the BEST Theorem to include these tours, linking their minimal length to the Pham index.

Findings

Multi-Eulerian tours exist in all strongly connected directed graphs.

The minimal length of such tours is bounded by the Pham index.

A generalized version of the BEST Theorem is presented.

Abstract

Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e)=tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of 'Eulerianness'.