1,175 research outputs found
Local connctedness and unicoherence at subcontinua
Let X be a continuum and Y a subcontinuum of X. The purpose of this paper is to investigate the relation between the conditions "X is unicoherent at Y" and "Y is unicoherent". We say that X is strangled by Y if the closure of each component of X Y intersects Y in one single point. We prove: If X is strangled by Y and Y is unicoherent then X is unicoherent at Y. We also prove the converse for a locally connected (not necessarily metric) continuum X
Colourful transversal theorems
We prove the colourful versions of three clasical transversal theorems: The Katchalski-Lewis Theorem "T(3) implies T-k", the "T(3) implies T" Theorem for well distributed sets, and the Goodmann-Pollack Transversal Theorem for hyperplanes
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
The Graphicahedron
The paper describes a construction of abstract polytopes from Cayley graphs
of symmetric groups. Given any connected graph G with p vertices and q edges,
we associate with G a Cayley graph of the symmetric group S_p and then
construct a vertex-transitive simple polytope of rank q, called the
graphicahedron, whose 1-skeleton (edge graph) is the Cayley graph. The
graphicahedron of a graph G is a generalization of the well-known
permutahedron; the latter is obtained when the graph is a path. We also discuss
symmetry properties of the graphicahedron and determine its structure when G is
small.Comment: 21 pages (European Journal of Combinatorics, to appear
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