## Chromatic number of associahedron ★★

Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood

\begin{conjecture} Associahedra have unbounded chromatic number. \end{conjecture}

## Edge-Colouring Geometric Complete Graphs ★★

\begin{question} What is the minimum number of colours such that every complete geometric graph on $n$ vertices has an edge colouring such that: \begin{itemize} \item[Variant A] crossing edges get distinct colours, \item[Variant B] disjoint edges get distinct colours, \item[Variant C] non-disjoint edges get distinct colours, \item[Variant D] non-crossing edges get distinct colours. \end{itemize} \end{question}

Keywords: geometric complete graph, colouring

## 3-Colourability of Arrangements of Great Circles ★★

Consider a set $S$ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $S$ has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
\begin{conjecture} Every arrangement graph of a set of great circles is $3$-colourable. \end{conjecture}