123 problems

Let $\mathbb{P}_4$ be the prefix-reversal graph on the permutations of $\{1,2,3,4\}$, and let $\gamma(\mathbb{P}_4)$ denote its orientable genus. A graph is a pretzel graph if it a…

Nonexistence conjecture for the displayed cover. For every finite planar cover

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…

Let ${\cal M}$ be a map, and denote the Euler characteristics of ${\cal M}$, $phial({\cal M})$, and $skew({\cal M})$ by $\chi({\cal M})$, $\chi_p({\cal M})$, and $\chi_s({\cal M})$…

Let $G$ be a connected finite simple graph. A finite planar cover of $G$ is a finite graph that covers $G$ in the graph-theoretic sense, with the covering graph planar. Negami's Pl…

Let $M(G)$ be a connected simple cubic map of girth $4$ embedded in a genus-realizing orientable surface, and suppose all its belts have lengths $\ell$ with $\ell\equiv 0\pmod 4$.…

Let $g>0$. A simple tiling $\mathcal T$ of a genus-$g$ surface has an associated incidence theorem over a division ring ${\mathrm D}$, and $n$ denotes the dimension of the ambient…

Let $\mathcal T$ be a simple tiling of a surface of positive genus, meaning that its underlying graph is $(n+1)$-degenerate, and let ${\mathrm D}$ be a division ring. The incidence…

A separating non-contractible cycle (SNCC) is a cycle in a triangulation that is separating and non-contractible. Let $t$ be a triangulation and let $C$ be an SNCC of $t$ of shorte…

Fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $g_n/n\rightarrow\theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces. A simple sepa…

A simple separating non-contractible cycle (SNCC) is a simple cycle in a triangulation that is both separating and non-contractible. Let $t$ be an arbitrary triangulation without l…

Let $\Sigma$ be a surface with Euler genus $g$, and let $G$ be a string graph in $\Sigma$. A string representation $\mathcal{C}$ of $G$ in $\Sigma$ assigns a non-self-intersecting…

Oriented strong embedding conjecture. Every $2$-connected graph has a strong embedding on some orientable surface.

Let $G$ be a graph of order $n$, let $\overline{G}$ be its complement, and let $\gamma(G)$ denote the genus of $G$. For $n\geq 11$, consider the lower bound … for…

Let $H$ be a graph with a self-gluing $d711: U_1 \to U_2$, and let $K$ be a valid minimal cut for this self-gluing. Construct the swapping $(H',\psi)=\operatorname{Sw}(H,d711,K)$ b…

Let $\operatorname{tg}(G)$ denote the minimum genus of an orientable surface on which the graph $G$ can be thrackled. Linear thrackle-genus conjecture. … The source has already est…

Let $S_g$ be a compact orientable surface of genus $g>0$, and let $G$ be a graph with $n$ vertices and $m$ edges. Cairns–Nikolayevsky's conjecture. If $G$ can be thrackled on $S_g$…

Let $S$ be a surface. Robertson's surface conjecture. There is a constant $r$ such that every $2$-edge-connected graph embedded in $S$ with representativity at least $r$ has a nowh…

Let $G$ be a $2$-edge-connected graph embedded in the torus. Robertson's conjecture. If the representativity of the embedding is at least $3$, then $G$ has a nowhere-zero $4$-flow.…

Let $G$ be a $2$-connected cubic graph embedded in the torus. Grünbaum's conjecture. If $G$ is not $3$-edge-colorable, then $G$ contains two edges on the same face of the embedding…

A face-transitive Klein bottle is a simplicial Klein bottle whose face-transitive symmetry property is understood in the sense of the paper's census. Face-transitive Klein bottle c…

For a simple drawing of $K_n$, let $\triangle(n)$ denote the minimum number of empty triangles among all such drawings. An empty triangle is a triangle induced by three vertices su…

A $k$-crossing family is a set of $k$ independent edges that cross pairwise. A simple drawing of $K_n$ is crossing-maximal when it has the maximum possible number $\binom{n}{4}$ of…

An h-convex drawing of $K_n$ is a drawing in which the vertices are in convex position with respect to the relevant hull structure. H-convex all-edges-crossed conjecture. For…