21,542 research outputs found
Upper and Lower Bounds on Long Dual-Paths in Line Arrangements
Given a line arrangement with lines, we show that there exists a
path of length in the dual graph of formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with blue and red lines
with no alternating path longer than . Further, we show that any line
arrangement with lines has a coloring such that it has an alternating path
of length . Our results also hold for pseudoline
arrangements.Comment: 19 page
Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem
We consider arrangements of axis-aligned rectangles in the plane. A geometric
arrangement specifies the coordinates of all rectangles, while a combinatorial
arrangement specifies only the respective intersection type in which each pair
of rectangles intersects. First, we investigate combinatorial contact
arrangements, i.e., arrangements of interior-disjoint rectangles, with a
triangle-free intersection graph. We show that such rectangle arrangements are
in bijection with the 4-orientations of an underlying planar multigraph and
prove that there is a corresponding geometric rectangle contact arrangement.
Moreover, we prove that every triangle-free planar graph is the contact graph
of such an arrangement. Secondly, we introduce the question whether a given
rectangle arrangement has a combinatorially equivalent square arrangement. In
addition to some necessary conditions and counterexamples, we show that
rectangle arrangements pierced by a horizontal line are squarable under certain
sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
Valid Orderings of Real Hyperplane Arrangements
Given a real finite hyperplane arrangement A and a point p not on any of the
hyperplanes, we define an arrangement vo(A,p), called the *valid order
arrangement*, whose regions correspond to the different orders in which a line
through p can cross the hyperplanes in A. If A is the set of affine spans of
the facets of a convex polytope P and p lies in the interior of P, then the
valid orderings with respect to p are just the line shellings of p where the
shelling line contains p. When p is sufficiently generic, the intersection
lattice of vo(A,p) is the *Dilworth truncation* of the semicone of A. Various
applications and examples are given. For instance, we determine the maximum
number of line shellings of a d-polytope with m facets when the shelling line
contains a fixed point p. If P is the order polytope of a poset, then the sets
of facets visible from a point involve a generalization of chromatic
polynomials related to list colorings.Comment: 15 pages, 2 figure
Enumerating Colorings, Tensions and Flows in Cell Complexes
We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex , generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in for
some ) or integral (with values in ). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory
Series
Relative cohomology of bi-arrangements
A bi-arrangement of hyperplanes in a complex affine space is the data of two
sets of hyperplanes along with a coloring information on the strata. To such a
bi-arrangement, one naturally associates a relative cohomology group, that we
call its motive. The motivation for studying such relative cohomology groups
comes from the notion of motivic period. More generally, we suggest the
systematic study of the motive of a bi-arrangement of hypersurfaces in a
complex manifold. We provide combinatorial and cohomological tools to compute
the structure of these motives. Our main object is the Orlik-Solomon bi-complex
of a bi-arrangement, which generalizes the Orlik-Solomon algebra of an
arrangement. Loosely speaking, our main result states that "the motive of an
exact bi-arrangement is computed by its Orlik-Solomon bi-complex", which
generalizes classical facts involving the Orlik-Solomon algebra of an
arrangement. We show how this formalism allows us to explicitly compute motives
arising from the study of multiple zeta values and sketch a more general
application to periods of mixed Tate motives.Comment: 43 pages; minor correction
Inside-Out Polytopes
We present a common generalization of counting lattice points in rational
polytopes and the enumeration of proper graph colorings, nowhere-zero flows on
graphs, magic squares and graphs, antimagic squares and graphs, compositions of
an integer whose parts are partially distinct, and generalized latin squares.
Our method is to generalize Ehrhart's theory of lattice-point counting to a
convex polytope dissected by a hyperplane arrangement. We particularly develop
the applications to graph and signed-graph coloring, compositions of an
integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat
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