Drawings of complete graphs in the projective plane Article uri icon

abstract

  • Hill%27s Conjecture states that the crossing number (Formula presented.) of the complete graph (Formula presented.) in the plane (equivalently, the sphere) is (Formula presented.). Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely (Formula presented.), thus matching asymptotically the conjectured value of (Formula presented.). Let (Formula presented.) denote the crossing number of a graph (Formula presented.) in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of (Formula presented.) is (Formula presented.). In analogy with the relation of Moon%27s result to Hill%27s conjecture, Elkies asked if (Formula presented.). We construct drawings of (Formula presented.) in the projective plane that disprove this. © 2021 Wiley Periodicals LLC
  • Hill's Conjecture states that the crossing number (Formula presented.) of the complete graph (Formula presented.) in the plane (equivalently, the sphere) is (Formula presented.). Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely (Formula presented.), thus matching asymptotically the conjectured value of (Formula presented.). Let (Formula presented.) denote the crossing number of a graph (Formula presented.) in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of (Formula presented.) is (Formula presented.). In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if (Formula presented.). We construct drawings of (Formula presented.) in the projective plane that disprove this. © 2021 Wiley Periodicals LLC

publication date

  • 2021-01-01