Long directed paths in Eulerian digraphs

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 $\mathrm{\Omega }(d)$. The best known lower bound for this problem is $\mathrm{\Omega }(d^{1\mathrm{/}2})$ 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 $\mathrm{\Omega }(d^{1\mathrm{/}2+1\mathrm{/}40})$ from any vertex of a connected Eulerian digraph.