15,258 research outputs found
Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
Flips of diagonals in colored triangle-free triangulations of a convex
polygon are interpreted as moves between two adjacent chambers in a certain
graphic hyperplane arrangement. Properties of geodesics in the associated flip
graph are deduced. In particular, it is shown that: (1) every diagonal is
flipped exactly once in a geodesic between distinguished pairs of antipodes;
(2) the number of geodesics between these antipodes is equal to twice the
number of Young tableaux of a truncated shifted staircase shape.Comment: figure added, plus several minor change
Tilings of the Sphere by Edge Congruent Pentagons
We study edge-to-edge tilings of the sphere by edge congruent pentagons,
under the assumption that there are tiles with all vertices having degree 3. We
develop the technique of neighborhood tilings and apply the technique to
completely classify edge congruent earth map tilings.Comment: 36 pages, 34 figure
Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
In this work we consider triangulations of point sets in the Euclidean plane,
i.e., maximal straight-line crossing-free graphs on a finite set of points.
Given a triangulation of a point set, an edge flip is the operation of removing
one edge and adding another one, such that the resulting graph is again a
triangulation. Flips are a major way of locally transforming triangular meshes.
We show that, given a point set in the Euclidean plane and two
triangulations and of , it is an APX-hard problem to minimize
the number of edge flips to transform to .Comment: A previous version only showed NP-completeness of the corresponding
decision problem. The current version is the one of the accepted manuscrip
Sparse Kneser graphs are Hamiltonian
For integers and , the Kneser graph is the
graph whose vertices are the -element subsets of and whose
edges connect pairs of subsets that are disjoint. The Kneser graphs of the form
are also known as the odd graphs. We settle an old problem due to
Meredith, Lloyd, and Biggs from the 1970s, proving that for every ,
the odd graph has a Hamilton cycle. This and a known conditional
result due to Johnson imply that all Kneser graphs of the form
with and have a Hamilton cycle. We also prove that
has at least distinct Hamilton cycles for .
Our proofs are based on a reduction of the Hamiltonicity problem in the odd
graph to the problem of finding a spanning tree in a suitably defined
hypergraph on Dyck words
The Canada Day Theorem
The Canada Day Theorem is an identity involving sums of minors
of an arbitrary symmetric matrix. It was discovered as a
by-product of the work on so-called peakon solutions of an integrable nonlinear
partial differential equation proposed by V. Novikov. Here we present another
proof of this theorem, which explains the underlying mechanism in terms of the
orbits of a certain abelian group action on the set of all -edge matchings
of the complete bipartite graph .Comment: 16 pages. pdfLaTeX + AMS packages + TikZ. Fixed a hyperlink problem
and a few typo
- …