16 research outputs found
Note on the Irreducible Triangulations of the Klein Bottle
We give the complete list of the 29 irreducible triangulations of the Klein
bottle. We show how the construction of Lawrencenko and Negami, which listed
only 25 such irreducible triangulations, can be modified at two points to
produce the 4 additional irreducible triangulations of the Klein bottle.Comment: 10 pages, 8 figures, submitted to Journal of Combinatorial Theory,
Series B. Section 3 expande
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
Irreducible triangulations of the once-punctured torus
A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the once-punctured torus, consisting of exactly 297 non-isomorphic triangulations.Plan Andaluz de Investigación (Junta de AndalucÃa)Ministerio de Ciencia e Innovació
The maximum number of cliques in a graph embedded in a surface
This paper studies the following question: Given a surface and an
integer , what is the maximum number of cliques in an -vertex graph
embeddable in ? We characterise the extremal graphs for this question,
and prove that the answer is between and , where is the maximum integer such that the
complete graph embeds in . For the surfaces ,
, , , ,
and we establish an exact answer
Surface realization with the intersection edge functional
Deciding realizability of a given polyhedral map on a (compact, connected)
surface belongs to the hard problems in discrete geometry, from the
theoretical, the algorithmic, and the practical point of view.
In this paper, we present a heuristic algorithm for the realization of
simplicial maps, based on the intersection edge functional. The heuristic was
used to find geometric realizations in R^3 for all vertex-minimal
triangulations of the orientable surfaces of genus g=3 and g=4. Moreover, for
the first time, examples of simplicial polyhedra in R^3 of genus 5 with 12
vertices were obtained.Comment: 22 pages, 11 figures, various minor revisions, to appear in
Experimental Mathematic