Crossing and intersecting families of geometric graphs on point sets
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373_2023_Article_2734.pdf
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Author(s)
Álvarez-Rebollar, J. L.
Cravioto-Lagos, J.
Marín, N.
Solé-Pi, O.
Urrutia, J.
Date Issued
January 25, 2024
Publisher
Springer Japan
Citation
Graphs and Combinatorics. 2024 Jan 25;40(1):17
Version
Final published version
Abstract
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
$$n^c$$
n
c
mutually-crossing 2-paths (each of which is the result of deleting an edge from one of the triangles) and provide an example that implies that c cannot be taken to be larger than 2/3. Then, for every n we determine the maximum number of crossings that a Hamiltonian cycle on a set of n points might have, and give examples achieving this bound. 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
$$\lfloor n/4 \rfloor $$
⌊
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 (Acta Math Hung 15(2):301–311, 2019,
https://doi.org/10.1007/s10474-018-0880-1
), namely, that any set S of n points in general position admits a family of intersecting triangles with a quadratic number of elements. For points in convex position we prove that any set of 3n points in convex position contains a family with at least
$$n^2$$
n
2
intersecting triangles.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00373-023-02734-9
