5 problems

Cubic-cage conjecture. For infinitely many integers $g$, there exists a cubic graph $G$ such that

Cycle-separating cut conjecture. For each $(k,g)$-cage $G$, every cycle-separating edge-cut of size $(k-2)g$ in $G$ separates a $g$-cycle.

Cyclic edge-connectivity conjecture. Every $(k,g)$-cage is cyclically $(k-2)g$-edge-connected.

Let a $(d,g)$-cage $G$ be a graph of degree $d$ and girth $g$, and let $\operatorname{mp}(G)$ denote its monophonic position number. Thomas's conjecture. For sufficiently large $g$…

A $(d,g)$-cage is a graph of minimum possible order among graphs with degree $d$ and girth $g$, and ${\rm mp}(G)$ denotes the monophonic position number of $G$. The cage monophonic…