4 problems

Katona–Xiao conjecture.

Let $G$ be a connected Eulerian directed graph with average degree at least $d$, and let a directed path be a sequence of consistently oriented edges. Huang–Ma–Shapira–Sudakov–Yust…

Let $Q_n$ be the $n$-dimensional hypercube, and let $G$ be a subgraph of $Q_n$ with average degree at least $d$. The hypercube average-degree path conjecture. $G$ contains a path o…

Let $C_3^n$ be the $n$-dimensional discrete torus, and let $G$ be a subgraph of $C_3^n$ with minimum degree at least $d$. The discrete-torus path-length conjecture. $G$ must contai…