Note on Long Paths in Eulerian Digraphs

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.