Disjoint edges in geometric graphs (Q5925578)

This paper shows that in a geometric graph with \(n\) vertices and \(e\) edges there are at least \(\frac{n}{2}\binom{2e/n}{3}\) pairs of disjoint edges provided that \(2e\ge n\) and all the vertices of the graph are pointed. Moreover, it is proved 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. Some open problems and conjectures are also mentioned in the paper.