Abstract
A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the convex hull of its neighbours. We show that for a geometric graph with \(n\) vertices and \(e\) edges there are at least \(\frac{n}{2}\left(\begin{array}{cc}2e/n\\3\end{array}\right)\) pairs of disjoint edges provided that \(2e\ge n\) and all the vertices of the graph are pointed. Besides, we prove that if any edge of a geometric graph with \(n\) vertices is disjoint from at most \(m\) edges, then the number of edges of this graph does not exceed \(n(\sqrt{1+8m}+3)/4\) provided that \(n\) is sufficiently large. These two results are tight for an infinite family of graphs.
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The study of the second author was funded by Russian Science Foundation (Grant no. 21-71-10092). The second author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.
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Chernega, N., Polyanskii, A. & Sadykov, R. Disjoint Edges in Geometric Graphs. Graphs and Combinatorics 38, 162 (2022). https://doi.org/10.1007/s00373-022-02563-2
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DOI: https://doi.org/10.1007/s00373-022-02563-2