Note on Long Paths in Eulerian Digraphs

Charlotte Knierim,Maxime Larcher,Anders Martinsson

doi:10.37236/10297

Abstract

Long paths and cycles in Eulerian digraphs have received a lot of attention recently. In this short note, we show how to use methods from [Knierim, Larcher, Martinsson, Noever, JCTB 148:125--148] to find paths of length $d/(\log d+1)$ in Eulerian digraphs with average degree $d$, improving the recent result of $\Omega(d^{1/2+1/40})$. Our result is optimal up to at most a logarithmic factor.

Highlights

One of the fundamental questions of extremal combinatorics is to determine how ‘large’ a graph G may be before it needs to contain certain graphs as subgraphs
A famous result in this topic is a theorem of Erdos and Gallai [3], which states that any graph of average degree d contains a path of length linear in d
Bollobas and Scott [1] conjectured that all Eulerian digraphs with average degree1 d contain a path of length Ω(d)

Summary

Introduction

One of the fundamental questions of extremal combinatorics is to determine how ‘large’ a graph G may be before it needs to contain certain graphs as subgraphs. A famous result in this topic is a theorem of Erdos and Gallai [3], which states that any graph of average degree d contains a path N 2 in which all edges are oriented in the same direction — which have high average degree but only paths of length 1.

Results

Conclusion