15 problems

Domination conjecture for non-spherical surfaces. If $n$ is sufficiently large, then

Conjecture on diagonal-flip stabilization. The only surfaces $S$ for which

Conjecture on nonseparating cycle types. If $g \geq 3$ is odd, every triangulation of $N_g$ has a nonseparating cycle that is one-sided and orientable-leaving, one that is one-side…

Nonorientable analogue of Thomassen's conjecture. Every such triangulation contains an NSC such that the two surfaces separated by the NSC have Euler genera $h$ and $g-h$, respecti…

Barnette's conjecture. Every triangulation of a surface with genus greater than $1$ has an NSC.

Characteristic-zero exterior shifting conjecture.

Lex-segment prevalence conjecture. As $n$ tends to infinity,

Lexicographic edge conjecture. If $\sigma<_{\mathrm{lex}}\sigma'$, then

Bulavka–Nevo–Peled conjecture. The triangle $\{1,3,n\}$ belongs to $\Delta(K)$ for every triangulation $K$ of a compact surface without boundary.

Let $G$ be a triangulation of a surface, with $n$ vertices, and let $amma(G)$ denote its domination number. Domination-number lower-bound conjecture. The tight lower bound for…

Let $G$ be a graph with a triangulation $T$ of a surface $S$ as a spanning subgraph. A graph is globally rigid when its generic realization is uniquely determined, up to Euclidean…

Infinitude conjecture. For every $d\geq 10$, there are infinitely many geodesic self-dual surfaces.

Leading-order coefficient conjecture. We conjecture that $b_1=-2/5$ for orientable and $b_1=-1/5$ for non-orientable surface triangulations.

Intersection-matrix characterization conjecture. If $S$ and $T$ have the same intersection matrix, then the triangulations of $S$ and $T$ are isomorphic.

Let $(\Sigma,\mathbb{M})$ be a positive-genus surface with empty boundary and exactly two punctures, and let $\tau$ be a tagged triangulation of $(\Sigma,\mathbb{M})$. Write…