The crossing number of Cm × Cn is as conjectured for n ≥ m(m + 1)

  • Authors:

  • Lev Yu. Glebsky;Gelasio Salazar

  • Affiliations:

  • Department of Mechanics and Mathematics, Nizhny Novgorod State University, Russia;IICO–UASLP, Av. Karakorum 1470, Lomas 4ta. Seccion, San Luis Potosi SLP, Mexico 78210

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2004

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Abstract

It has been long conjectured that the crossing number of Cm × Cn is (m-2)n, for all m, n such that n ≥  m ≥  3. In this paper, it is shown that if n ≥  m(m + 1) and m ≥  3, then this conjecture holds. That is, the crossing number of Cm × Cn is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by Adamsson and Richter. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 53–72, 2004