In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive distinct colours in the other colouring. A random geometric graph $RG(n,r)$ is a graph constructed by randomly placing $n$ vertices in the unit square and connecting two vertices with an edge if and only if their distance is less than the threshold $r$. We show first that random geometric graphs with $r>n^{-α}$, where $0\leq α\leq\frac{1}{4}$, have an orthogonal colouring using $n^{1-2α}(1+o(1))$ colours with high probability. Then, we show for an infinite number of values of $n$, random geometric graphs with threshold $r<cn^{-\frac{1}{4}}$, $c<1$, have an optimal orthogonal colouring with high probability. We obtain both of these results by constructing orthogonal colourings of the clique grid graph.
Geometric difference families and related variable-weight geometric orthogonal codes
School of Mathematical Sciences, Hebei Normal University（河北师范大学数学科学学院）；College of Basic Education, Beijing Institute of Economics and Management（北京经济管理职业学院基础教育部）；Hebei Research Center of the Basic Discipline Pure Mathematics（河北省基础学科纯数学研究中心）
