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An Improved Bound for the Crossing Number of C m C n : a Self–Contained Proof Using Mostly Combinatorial Arguments

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Abstract.

We present some surprisingly elementary arguments to prove that for every ε>0, if m is sufficiently large, then the crossing number of the Cartesian product C m ×C n is at least (0.8−ε)m n, for every n≥m. The self–contained proof we give involves only one (rather elementary) geometrical result. The rest of the proof involves purely combinatorial arguments.

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Authors and Affiliations

  1. IICO–UASLP, Av. Karakorum 1470, Lomas 4ta. Seccion, San Luis Potosi, SLP 78210, Mexico

    Gelasio Salazar & Edgardo Ugalde

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  1. Gelasio Salazar
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  2. Edgardo Ugalde
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Correspondence to Gelasio Salazar.

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Supported by CONA-CYT (Grant J32168E) and by FAI-UASLP

Final version received: July 14, 2003

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Salazar, G., Ugalde, E. An Improved Bound for the Crossing Number of C m C n : a Self–Contained Proof Using Mostly Combinatorial Arguments. Graphs and Combinatorics 20, 247–253 (2004). https://doi.org/10.1007/s00373-003-0549-5

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  • DOI: https://doi.org/10.1007/s00373-003-0549-5

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