Note on Long Directed Cycles in Eulerian Digraphs

Huang, Ma, Shapira, Sudakov and Yuster (Comb. Prob. Comput. 2013) proved that every Eulerian digraph of average out-degree $d$ has a directed cycle of length at least $\sqrt{d}\mathrm{.}$ We improve the lower bound from $\sqrt{d}$ to $\sqrt{2d}-3\mathrm{/}2.$