Long directed paths in Eulerian digraphs

TL;DR

This paper improves the lower bound on the length of directed paths in Eulerian digraphs, showing that from any vertex in a connected Eulerian digraph, one can find a path of length significantly longer than the previous  bound, advancing understanding of graph connectivity.

Contribution

The authors establish a new lower bound of +1/40 for the length of directed paths in connected Eulerian digraphs, surpassing the previous  bound.

Findings

Longer directed paths can be found in Eulerian digraphs than previously known.

The new bound is +1/40, breaking the  barrier.

Results apply to any starting vertex in a connected Eulerian digraph.

Abstract

An old conjecture of Bollob\'as and Scott asserts that every Eulerian directed graph with average degree $d$ contains a directed cycle of length at least $\Omega(d)$. The best known lower bound for this problem is $\Omega(d^{1/2})$ by Huang, Ma, Shapira, Sudakov and Yuster. They asked whether this estimate can be improved at least for directed paths instead of cycles and whether one can find a long path starting from any vertex if the host digraph is connected. In this paper we break the $\sqrt{d}$ barrier, showing how to find a path of length $\Omega(d^{1/2+1/40})$ from any vertex of a connected Eulerian digraph.