115 research outputs found
Realizing the chromatic numbers and orders of spinal quadrangulations of surfaces
A method is suggested for construction of quadrangulations of the closed
orientable surface with given genus g and either (1) with given chromatic
number or (2) with given order allowed by the genus g. In particular, N.
Hartsfield and G. Ringel's results [Minimal quadrangulations of orientable
surfaces, J. Combin. Theory, Series B 46 (1989) 84-95] are generalized by way
of generating new minimal quadrangulations of infinitely many other genera.Comment: 6 pages. This version is only slightly different from the original
version submitted on 8 Jul 2012: the author's affiliation has been changed
and the presentation has been slightly improve
A topological classification of convex bodies
The shape of homogeneous, generic, smooth convex bodies as described by the
Euclidean distance with nondegenerate critical points, measured from the center
of mass represents a rather restricted class M_C of Morse-Smale functions on
S^2. Here we show that even M_C exhibits the complexity known for general
Morse-Smale functions on S^2 by exhausting all combinatorial possibilities:
every 2-colored quadrangulation of the sphere is isomorphic to a suitably
represented Morse-Smale complex associated with a function in M_C (and vice
versa). We prove our claim by an inductive algorithm, starting from the path
graph P_2 and generating convex bodies corresponding to quadrangulations with
increasing number of vertices by performing each combinatorially possible
vertex splitting by a convexity-preserving local manipulation of the surface.
Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist,
this algorithm not only proves our claim but also generalizes the known
classification scheme in [36]. Our expansion algorithm is essentially the dual
procedure to the algorithm presented by Edelsbrunner et al. in [21], producing
a hierarchy of increasingly coarse Morse-Smale complexes. We point out
applications to pebble shapes.Comment: 25 pages, 10 figure
Linear Complexity Hexahedral Mesh Generation
We show that any polyhedron forming a topological ball with an even number of
quadrilateral sides can be partitioned into O(n) topological cubes, meeting
face to face. The result generalizes to non-simply-connected polyhedra
satisfying an additional bipartiteness condition. The same techniques can also
be used to reduce the geometric version of the hexahedral mesh generation
problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at
the 12th ACM Symp. on Computational Geometry. This is the final version, and
will appear in a special issue of Computational Geometry: Theory and
Applications for papers from SCG '9
Equivelar and d-Covered Triangulations of Surfaces. I
We survey basic properties and bounds for -equivelar and -covered
triangulations of closed surfaces. Included in the survey is a list of the
known sources for -equivelar and -covered triangulations. We identify all
orientable and non-orientable surfaces of Euler characteristic
which admit non-neighborly -equivelar triangulations
with equality in the upper bound
. These
examples give rise to -covered triangulations with equality in the upper
bound . A
generalization of Ringel's cyclic series of neighborly
orientable triangulations to a two-parameter family of cyclic orientable
triangulations , , , is the main result of this
paper. In particular, the two infinite subseries and
, , provide non-neighborly examples with equality for
the upper bound for as well as derived examples with equality for the upper
bound for .Comment: 21 pages, 4 figure
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