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arXiv · 2303.08211

Orthogonal Colourings of Random Geometric Graphs

Jeannette Janssen·Kyle MacKeigan
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Abstract

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 $r>n^{-\alpha}$, where $0\leq \alpha \leq\frac{1}{4}$, have an orthogonal colouring using $n^{1-2\alpha}(1+o(1))$ 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.

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Jeannette Janssen, Kyle MacKeigan. 2023-03-14. Orthogonal Colourings of Random Geometric Graphs. https://arxiv.org/abs/2303.08211

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