Multi-Eulerian Tours of Directed Graphs

Matthew Farrell,Lionel Levine

doi:10.37236/5588

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'.

Highlights

In the following G = (V, E) denotes a finite directed graph, with loops and multiple edges permitted
We show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of ‘Eulerianness’
The purpose of this note is to generalize the notion of Eulerian tour and the BEST theorem to any finite, strongly connected graph G

Summary

Introduction

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 ). The purpose of this note is to generalize the notion of Eulerian tour and the BEST theorem to any finite, strongly connected graph G. Note that existence of a π-Eulerian tour implies that G is strongly connected: for each v, w ∈ V there are directed paths from v to w and from w to v.

Results

Conclusion