2 problems

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 $G$…

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.