4 problems

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

Minimal fullerene thimble conjecture. For odd or even $c\geq 5$, respectively, the stated $(5c-5)$- or $(5c-6)$-vertex $c$-thimble is minimal.

Singleton-face conjecture. The face sizes satisfy $c=b$; equivalently, the map is an $(\{a,b\},k)$-sphere with $p_b=2$.

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