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The Beginnings of Geometric Graph Theory

  • Chapter
Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 25))

Abstract

Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straight-line edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal paper of Paul Erdős from 1946, we give a biased survey of Turán-type questions in the theory of geometric and topological graphs. What is the maximum number of edges that a geometric or topological graph of n vertices can have if it contains no forbidden subconfiguration of a certain type? We put special emphasis on open problems raised by Erdős or directly motivated by his work.

“… to ask the right question and to ask it of the right person.” (Richard Guy)

Supported by NSF Grant CCF-08-30272, by OTKA under EUROGIGA projects GraDR and ComPoSe 10-EuroGIGA-OP-003, and by Swiss National Science Foundation Grants 200020-144531 and 200021-137574.

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Authors and Affiliations

  1. EPFL, Station 8, CH-1015, Lausanne, Switzerland

    János Pach

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13-15, H-1053, Hungary

    János Pach

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  1. János Pach
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  1. Department of Computer Science, Eötvös Lóránd University, Pázmány P. sétány 1/c, Budapest, 1117, Hungary

    László Lovász

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Imre Z. Ruzsa  & Vera T. Sós  & 

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Pach, J. (2013). The Beginnings of Geometric Graph Theory. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_17

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