Note on Long Paths in Eulerian Digraphs

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 $d\mathrm{/}(\log d+1)$ in Eulerian digraphs with average degree $d$, improving the recent result of $\mathrm{\Omega }(d^{1\mathrm{/}2+1\mathrm{/}40})$. Our result is optimal up to at most a logarithmic factor.