947 research outputs found
On long cycles through four prescribed vertices of a polyhedral graph
For a 3-connected planar graph G with circumference c ≥ 44 it is proved that G has a cycle of length at least [1/36]c+[20/3] through any four vertices of G
On short cycles through prescribed vertices of a polyhedral graph
Guaranteed upper bounds on the length of a shortest cycle through k ≤ 5 prescribed vertices of a polyhedral graph or plane triangulation are proved
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
Combinatorics of tropical Hurwitz cycles
We study properties of the tropical double Hurwitz loci defined by Bertram,
Cavalieri and Markwig. We show that all such loci are connected in codimension
one. If we mark preimages of simple ramification points, then for a generic
choice of such points the resulting cycles are weakly irreducible, i.e. an
integer multiple of an irreducible cycle. We study how Hurwitz cycles can be
written as divisors of rational functions and show that they are numerically
equivalent to a tropical version of a representation as a sum of boundary
divisors. The results and counterexamples in this paper were obtained with the
help of a-tint, an extension for polymake for tropical intersection theory.Comment: 29 pages, 16 figures. Minor revisions, to appear in Journal of
Algebraic Combinatoric
Tropical Hurwitz Numbers
Hurwitz numbers count genus g, degree d covers of the projective line with
fixed branch locus. This equals the degree of a natural branch map defined on
the Hurwitz space. In tropical geometry, algebraic curves are replaced by
certain piece-wise linear objects called tropical curves. This paper develops a
tropical counterpart of the branch map and shows that its degree recovers
classical Hurwitz numbers.Comment: Published in Journal of Algebraic Combinatorics, Volume 32, Number 2
/ September, 2010. Added section on genus zero piecewise polynomiality.
Removed paragraph on psi classe
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