Dual Circumference and Collinear Sets

We show that, if a $n$-vertex triangulation $T$ of maximum degree $Δ$ has a dual that contains a cycle of length $\ell$, then $T$ has a non-crossing straight-line drawing in which some \emph{collinear set} of $Ω(\ell/Δ^4)$ vertices lie on a line. Using the current lower bounds on the length of longest cycles in 3-regular 3-connected graphs, this implies that every $n$-vertex planar graph of maximum degree $Δ$ has a collinear set of size $Ω(n^{0.8}/Δ^4)$. Very recently, Dujmović et. al. (SODA 2019) showed that, if $S$ is a collinear set in a triangulation $T$ then, for any point set $X\subset\mathbb{R}^2$ with $|X|=|S|$, $T$ has a non-crossing straight-line drawing in which the vertices of $S$ are drawn on the points in $X$. Because of this, collinear sets have numerous applications in graph drawing and related areas.

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