Crossing and intersecting families of geometric graphs on point sets
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Let S be a set of n points in the plane in general position. Two line segments connecting pairs of points of S cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in S cross if there are two edges, one from each graph, which cross. A set of vertex disjoint geometric graphs with vertices in S is called mutually crossing if any two of them cross. We show that there exists a constant c such that from any family of n mutually crossing triangles, one can always obtain a family of at least n^c mutually crossing 2-paths (each of which is the result of deleting an edge from one of the triangles) and then provide an example that implies that c cannot be taken to be larger than 2/3. For every n we determine the maximum number of crossings that a Hamiltonian cycle on a set of n points might have. Next, we construct a point set whose longest perfect matching contains no crossings. We also consider edges consisting of a horizontal and a vertical line segment joining pairs of points of S, which we call elbows, and prove that in any point set S there exists a family of ⌊ n/4 ⌋ vertex disjoint mutually crossing elbows. Additionally, we show a point set that admits no more than n/3 mutually crossing elbows. Finally we study intersecting families of graphs, which are not necessarily vertex disjoint. A set of edge disjoint graphs with vertices in S is called an intersecting family if for any two graphs in the set we can choose an edge in each of them such that they cross. We prove a conjecture by Lara and Rubio-Montiel, namely, that any set S of n points in general position admits a family of intersecting triangles with a quadratic number of elements. Some other results are obtained throughout this work.
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