6 problems

General $k$-degenerate-cut conjecture. Every graph of sufficiently large order $n$ with fewer than

Kaneko's conjecture. Every graph of order $n$ with fewer than $3n-6$ edges has a forest cut.

Bipartite-cut conjecture. If a graph $G$ with $n\geqslant 4$ vertices has fewer than $3n-6$ edges, then $G$ admits a bipartite cut.

$\alpha$-FC conjecture. If $G$ has no forest cut, then

Let $G$ be a finite, simple, undirected graph of order $n$, and call a vertex set a forest cut if it is a vertex cut whose induced subgraph is a forest. Forest-cut conjecture. If…

Peripheral vertex-cut conjecture. If $G$ is $k$-connected and its periphery has cardinality at least $k$, then $G$ contains a set $S$ of $k$ peripheral vertices such that $G-S$ is…