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The Crossing Number of the Cone of a Graph

  • Carlos A. AlfaroEmail author
  • Alan Arroyo
  • Marek Derňár
  • Bojan Mohar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

Motivated by a problem asked by Richter and by the long standing Harary-Hill conjecture, we study the relation between the crossing number of a graph G and the crossing number of its cone CG, the graph obtained from G by adding a new vertex adjacent to all the vertices in G. Simple examples show that the difference \(cr(CG)-cr(G)\) can be arbitrarily large for any fixed \(k=cr(G)\). In this work, we are interested in finding the smallest possible difference, that is, for each non-negative integer k, find the smallest f(k) for which there exists a graph with crossing number at least k and cone with crossing number f(k). For small values of k, we give exact values of f(k) when the problem is restricted to simple graphs, and show that \(f(k)=k+\varTheta (\sqrt{k})\) when multiple edges are allowed.

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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Carlos A. Alfaro
    • 1
    Email author
  • Alan Arroyo
    • 2
  • Marek Derňár
    • 3
  • Bojan Mohar
    • 4
  1. 1.Banco de MéxicoMexico CityMexico
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  3. 3.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic
  4. 4.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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