4 problems

The $\frac{5}{4}$-tough spanning 2-trail conjecture. Any $\frac{5}{4}$-tough $2K_2$-free graph with at least three vertices has a spanning 2-trail.

Let $G$ be a $k$-edge-connected graph, meaning every edge cut of $G$ has size at least $k$, with $k\geq 4$. A spanning closed trail is a closed trail containing every vertex of $G$…

Bounded-trail conjecture. For every integer $\chi$, there is a positive integer $k_\chi$ such that every $4$-connected graph which embeds on a surface with Euler characteristic…

Mou and Pasechnik's conjecture. Every $\frac{3}{2}$-tough $2K_2$-free graph with at least three vertices has a spanning 2-trail.