Note on Long Paths in Eulerian Digraphs

TL;DR

This paper improves the known bounds on the length of long paths in Eulerian digraphs by applying methods from recent research, achieving near-optimal results up to a logarithmic factor.

Contribution

It introduces a new approach to find longer paths in Eulerian digraphs, surpassing previous bounds and approaching optimality.

Findings

Paths of length d/(log d + 1) in Eulerian digraphs with average degree d

Improved lower bounds on long paths compared to previous results

Results are optimal up to a logarithmic factor

Abstract

Long Paths and Cycles in eulerian digraphs have gotten a lot of attention recently. In this short note, we show how to use methods from Knierim, Larcher, Martinsson, Noever (2021) 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.