# surface

## 5-local-tensions ★★

Author(s): DeVos

\begin{conjecture} There exists a fixed constant $c$ (probably $c=4$ suffices) so that every embedded (loopless) graph with edge-width $\ge c$ has a 5-local-tension. \end{conjecture}

## Drawing disconnected graphs on surfaces ★★

Author(s): DeVos; Mohar; Samal

\begin{conjecture} Let $G$ be the disjoint union of the graphs $G_1$ and $G_2$ and let $\Sigma$ be a surface. Is it true that every optimal drawing of $G$ on $\Sigma$ has the property that $G_1$ and $G_2$ are disjoint? \end{conjecture}

Keywords: crossing number; surface

## Grunbaum's Conjecture ★★★

Author(s): Grunbaum

\begin{conjecture} If $G$ is a simple loopless \Def[triangulation]{triangulation (topology)} of an \Def{orientable surface}, then the dual of $G$ is 3-edge-colorable. \end{conjecture}

## Consecutive non-orientable embedding obstructions ★★★

Author(s):

\begin{conjecture} Is there a graph $G$ that is a minor-minimal obstruction for two non-orientable surfaces? \end{conjecture}