13 problems

Isoperimetric optimality conjecture. The 120-cell has the least volume among all 4-polytopes of unit in-radius having 120 cells.

DDF's conjecture. For every even $n\geq 20$ with $n\ne 22$, there exists a tight $5_n$.

Let $C_N$ be the class of fullerene isomers with $N$ atoms. Assume $N\neq 20,24,26,28,34$, $N\neq 10s$, and $N\neq 6k-4$, where $k\geq 5$ and $s\geq 2$. A nanotubical fullerene gra…

Let $C_N$ be the class of fullerene isomers with $N$ atoms. Suppose $N=6k-4\neq 10s$, where $k\geq 5$ and $s\geq 2$. A nanotubical fullerene has a tubular part and two caps, and ca…

Let $C_N$ be the class of fullerene isomers with $N$ atoms, and let $N=10k$ with $k\geq 2$. A nanotubical fullerene has a tubular part and two caps; a cap of type (a) is one of the…

Let a fullerene graph be a graph of an arbitrary order, and let its Wiener index be the sum of graph distances over all unordered pairs of vertices. A nanotubical fullerene graph i…

Spectral characterization conjecture. (a) For all feasible $n\ne44$, two $C_n$-isomers are isomorphic if and only if they are cospectral with respect to $T_n$.

Chemical-stability conjecture. The family of icosahedral fullerenes has the highest chemical stability among all fullerenes with $n$ atoms.

Fullerene saturation-number conjecture. There is a constant $C$ such that

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.

Let $H$ be a fullerene graph with $n$ vertices, and let $b(H)$ be the minimum number of edges whose removal makes $H$ bipartite. Doš​lić–Vukićević explicit conjecture. … This is pr…

Let $H$ be a fullerene graph, let $V(H)$ denote its vertex set, and let $b(H)$ be the minimum number of edges whose removal makes $H$ bipartite. Doš​lić–Vukićević conjecture. They…