3 problems

Connected-graph contraction conjecture. Contracting any edge should reduce the mean subtree order by at least $\frac13$, with equality only for a path:

Let $G$ be a connected undirected graph, let $s$ be a vertex of $G$, and let $N$ be the set of neighbors of $s$, excluding $s$. For each nonempty subset $W\subseteq N$, let…

Quadratic average-degree conjecture. There exists a constant $c$ such that every finite $k$-connected graph of average degree at least $c k^2$ admits a $k$-contractible edge.