26 problems

A smoothly embedded projective plane in $S^4$ is knotted if it is not the standard unknotted embedding, and it is reducible if it admits an unknotted summand that is not a $2$-sphe…

Toroidal shallow-minor conjecture. There is no constant $k\in\mathbb N$ with the following property: every toroidal graph is a congestion-$k$ depth-$k$ minor of a planar graph.

Surface transduction-order conjecture. Let $\Sigma$ and $\Gamma$ be surfaces such that $\Sigma\not\preccurlyeq\Gamma$. Then

Let $\Sigma$ be a surface and let $g(\Sigma)$ denote its genus. Fix an integer $k$ for which there is a value $\rho_0=\rho_0(g(\Sigma))$ such that every $\rho_0$-locally planar gra…

Let $G$ be a graph embedded in a surface $\Sigma$, and let $g(\Sigma)$ denote the genus of $\Sigma$. A graph is $\rho$-locally planar if the relevant local planarity condition hold…

Jackson and Ringel's conjecture. For all $g\geq 1$, the bound in this inequality is tight:

Genus-square-root conjecture. If $G$ is a graph of genus $g$, then

Let $g$ be a genus function and let $\mathcal A^g$ be one of the labelled graph classes considered in the paper. Logarithmic-threshold growth-constant conjecture. If $c>0$ is a con…

Let $g$ be a genus function, and let $\mathcal A^g$ denote one of the graph classes considered in the paper, such as $\mathcal E^g$, $\mathcal O\mathcal E^g$,…

Let $\mathcal A^h$ be one of the graph classes considered in the paper, indexed by graphs on $[n]$ embeddable in surfaces of Euler genus at most $h$. Genus-increment growth-ratio c…

Let $\jump(g,r)$ be the maximum choice–chromatic gap among graphs embeddable on the orientable surface of genus $g$ with chromatic number $r$, and let $H(g)$ be the maximum chromat…

Let $\jump(g,r)$ denote the maximum of the difference between list-chromatic number and chromatic number over graphs embeddable on the orientable surface of genus $g$ with chromati…

Triangle-maximisation conjecture. If

Simultaneous clique-maximisation conjecture. If $C(G)=C(\Sigma,n)$, then for every $s\geqslant 0$,

Let $\mathbb{N}_3$ denote Dyck's surface, and let an Euler impure graph be an edge-maximal graph embeddable on a surface that does not triangulate that surface. For a graph $G$ and…

Weinstein-trisection characterization conjecture. The surface $\mathcal{K}$ is isotopic to a symplectic surface if and only if it is isotopic into transverse bridge position with r…

Let $G$ be a map on a genus $g\geq 1$ orientable surface. An edge, positive-vertex, positive-face angle labeling means a labeling with the edge condition and positive vertex and fa…

Maximal surface-graph conjecture. The sequence $G_n$ Benjamini--Schramm converges, and its limit coincides with the UIPT.

Angle-labeling characterization of essentially 3-connected maps. The map $M$ admits an edge, $\mathbb{N}^*$-vertex, $\mathbb{N}^*$-face angle labeling if and only if it is essentia…

Let $C_m$ be a cycle of even length $m\ge6$, let $P_n$ be a path of odd order $n\ge5$, and let $C_m\bowtie P_n$ denote their bow-tie graph. 3-extendability conjecture. For any even…

Let $G$ be a connected graph of orientable genus $h$ and non-orientable genus $k$. Let $b(G)$ be its bondage number and $\Delta(G)$ its maximum vertex degree. Let $c_h$ and $c'_k$…

Embedding conjecture. Every graph $G\in\mathcal{F}_{xy}^k$ embeds in $\mathbb{S}_k$.

Large-edgewidth boxicity conjecture. There exists an integer $e_\Sigma$ such that every graph $G$ embeddable on $\Sigma$ with edgewidth at least $e_\Sigma$ has boxicity at most thr…

Let $S$ be a fixed surface. For a graph $G$ embeddable on $S$ with a $\Sigma$-system, define … and let $\Delta(G;\Sigma)=\max_{v}|\Sigma(v)|$. The surface $\Sigma$-system list-colo…

DeVos–Mohar–Samal conjecture. For every optimal drawing of $G$ on $\Sigma$, the restrictions to $H$ and $K$ do not intersect.