Topological Graphs with No Large Grids

Authors:
János Pach;Rom Pinchasi;Micha Sharir;Géza Tóth
Affiliations:
City College, CUNY and Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA;Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA;School of Computer Science, Tel Aviv University, 69 978, Tel Aviv, MA, Israel and Courant Institute of Mathematiscal Sciences, New York University, New York, NY, 10012, USA;Rényi Institute, Hungarian Academy of Sciences, 69 978, Budapest, MA, Hungary
Venue:
Graphs and Combinatorics
Year:
2005

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Abstract

Let G be a topological graph with n vertices, i.e., a graph drawn in the plane with edges drawn as simple Jordan curves. It is shown that, for any constants k,l, there exists another constant C(k,l), such that if G has at least C(k,l)n edges, then it contains a k×l-gridlike configuration, that is, it contains k+l edges such that each of the first k edges crosses each of the last l edges. Moreover, one can require the first k edges to be incident to the same vertex.