2,044 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
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
Lattice Topological Field Theory on Non-Orientable Surfaces
The lattice definition of the two-dimensional topological quantum field
theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is
generalized to arbitrary (not necessarily orientable) compact surfaces. It is
shown that there is a one-to-one correspondence between real associative
-algebras and the topological state sum invariants defined on such surfaces.
The partition and -point functions on all two-dimensional surfaces
(connected sums of the Klein bottle or projective plane and -tori) are
defined and computed for arbitrary -algebras in general, and for the the
group ring of discrete groups , in particular.Comment: Corrected Latex file, 39 pages, 28 figures available upon reques
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
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