604 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
Colorful Associahedra and Cyclohedra
Every n-edge colored n-regular graph G naturally gives rise to a simple
abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic
to G. The paper describes colorful polytope versions of the associahedron and
cyclohedron. Like their classical counterparts, the colorful associahedron and
cyclohedron encode triangulations and flips, but now with the added feature
that the diagonals of the triangulations are colored and adjacency of
triangulations requires color preserving flips. The colorful associahedron and
cyclohedron are derived as colorful polytopes from the edge colored graph whose
vertices represent these triangulations and whose colors on edges represent the
colors of flipped diagonals.Comment: 21 pp, to appear in Journal Combinatorial Theory
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm
We prove that for the class of three-colorable triangulations of a closed
oriented surface, the degree of a four-coloring modulo 12 is an invariant under
Kempe changes. We use this general result to prove that for all triangulations
T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence
classes. This result implies in particular that the Wang-Swendsen-Kotecky
algorithm for the zero-temperature 4-state Potts antiferromagnet on these
triangulations T(3L,3M) of the torus is not ergodic.Comment: 37 pages (LaTeX2e). Includes tex file and 3 additional style files.
The tex file includes 14 figures using pstricks.sty. Minor changes. Version
published in J. Phys.
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