3 problems

Let $K$ be a wnp map, meaning a polyhedral map attaining equality in the lower bound $n\geq (k-1)^2$ for an $n$-vertex $\{k,k\}$-equivelar polyhedral map. Nonexistence conjecture.…

Fullerene disk existence conjecture. Except for $(c,v)=(1,42)$, $(3,24)$, and $(5,22)$, a fullerene $c$-disk with $c\geq 1$ and $v$ vertices exists if and only if $v$ is even and

The conjecture. For each surface $\mathbb{M}$, there exists a constant $c_{\mathbb{M}}$ such that, if