# Edge-Colouring Geometric Complete Graphs

 Importance: Medium ✭✭
 Subject: Geometry
 Keywords: geometric complete graph, colouring
 Posted by: David Wood on: October 19th, 2009

\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}

Let $P$ be a set of $n$ points in the plane with no three collinear. Draw a straight line-segment between each pair of points in $P$. We obtain the \emph{complete geometric graph} with vertex set $P$, denoted by $K_P$.

Two edges in $K_P$ are either: \begin{itemize} \item \emph{adjacent} if they have a vertex in common, \item \emph{crossing} if they intersect at a point in the interior of both edges. \item \emph{disjoint} if they do not intersect. \end{itemize}

Let $A(n)$, $B(n)$, $C(n)$ and $D(n)$ be the minimum number of colours for the four variants.

\textbf{Variant A:} Here each colour class is a plane subgraph. Since there are point sets for which $\frac{n}{2}$ edges are pairwise crossing, $A(n)\geq\frac{n}{2}$. For an upper bound, say $P=\{v_1,\dots,v_n\}$. Colour each edge $v_iv_j$ with $i \textbf{Conjecture.}$A(n)\leq (1-\epsilon)n$for some$\epsilon>0$. \textbf{Variant B:} Here edges receiving the same colour must intersect. So each colour class is a geometric thrackle. Since there are point sets for which$\frac{n}{2}$edges are pairwise disjoint,$B(n)\geq \frac{n}{2}$. The$(n-1)$-colouring given in Variant A also works here. So$B(n)\leq n-1$. \textbf{Conjecture.}$B(n)\leq (1-\epsilon)n$for some$\epsilon>0$. \textbf{Variant C:} Here each colour class is a plane matching. So each colour class has at most$\frac{n}{2}$edges, and thus at least$n-1$colours are always needed. Thus$C(n)\geq n-1$. Araujo [ADHNU] proved an upper bound of$C(n)\in O(n^{3/2})$. \textbf{Conjecture.}$C(n)\in O(n\log n)$. \textbf{Strong Conjecture.}$C(n)\in O(n)$. \textbf{Variant D:} (This variant was recently mentioned in [Mat].) Here edges receiving the same colour must cross. Each colour class is called a \emph{crossing family} [ADHNU]. Every edge in any triangulation of$P$requires its own colour. So if the convex hull of$P$has only three points, then at least$3n-6$colours are needed. Thus$D(n)\geq 3n-6$. \textbf{Conjecture.} A super-linear number of colours are always needed; i.e.,$\frac{D(n)}{n}\rightarrow\infty$as$n\rightarrow\infty$. A better lower bound is obtained by taking$P$in convex position. Then$\Theta(n\log n)$is the minimum number of colours [KK]. I am not aware of any non-trivial upper bound for arbitrary point sets$P\$.

## Bibliography

[ADHNU] G. Araujo, A. Dumitrescu, F. Hurtado, M. Noy, J. Urrutia, On the chromatic number of some geometric type Kneser graphs, \emph{Computational Geometry: Theory & Applications} 32(1):59–69, 2005 \MRhref{MR2155418}

[BHRW] Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood. Partitions of complete geometric graphs into plane trees, \emph{Computational Geometry: Theory & Applications} 34(2):116-125, 2006. \MRhref{MR2222887}

[AEGKKPS] B. Aronov, P. Erdos, W. Goddard, D.J. Kleitman, M. Klugerman, J. Pach, L.J. Schulman, Crossing families, \emph{Combinatorica} 14(2):127–134, 1994. \MRhref{MR1289067}

[KK] Alexandr Kostochka and Jan Kratochvil. Covering and coloring polygon-circle graphs, \emph{Discrete Math.} 163(1--3):299--305, 1997. \MRhref{MR1428585}

[Mat] Jiří Matoušek. Blocking visibility for points in general position. \emph{Discrete Comput. Geom.} 42(2):219--223, 2009. \MRhref{MR2519877}

* indicates original appearance(s) of problem.