Bollobás–Scott conjecture on long cycles in Eulerian digraphs

Let $G$ be a finite loopless directed graph, allowing both orientations of an edge, and write $d^+(v)$ and $d^-(v)$ for the outdegree and indegree of a vertex $v$. The graph is Eulerian if $d^+(v)=d^-(v)$ for every vertex $v$, with degree $d^+(v)=d^-(v)$ and average degree defined accordingly. Bollobás–Scott conjecture. If $G$ is Eulerian with average degree at least $d$, then $G$ contains a directed cycle of length at least $cd$ for some absolute constant $c>0$. This conjecture is still wide open: the best general lower bound mentioned in the paper is $(d^{1/2})$, with stronger bounds under additional maximum-degree assumptions, and the conjecture remains open even for simple graphs.