225 research outputs found
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
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Hypergraph coloring complexes
AbstractThe aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes–a purely combinatorial one and two geometric ones. It is shown, that most of the properties, which are known to be true for coloring complexes of graphs, break down in this more general setting, e.g., Cohen–Macaulayness and partitionability. Nevertheless, we are able to provide bounds for the f- and h-vectors of those complexes which yield new bounds on chromatic polynomials of hypergraphs. Moreover, though it is proven that the coloring complex of a hypergraph has a wedge decomposition, we provide an example showing that in general this decomposition is not homotopy equivalent to a wedge of spheres. In addition, we can completely characterize those hypergraphs whose coloring complex is connected
Nonclassical stochastic flows and continuous products
Contrary to the classical wisdom, processes with independent values (defined
properly) are much more diverse than white noise combined with Poisson point
processes, and product systems are much more diverse than Fock spaces.
This text is a survey of recent progress in constructing and investigating
nonclassical stochastic flows and continuous products of probability spaces and
Hilbert spaces.Comment: A survey, 126 pages. Version 3 (final): former Question 9d4 is
solved; 8a1 reformulated. Ref [41] added. For readability, sections are
reordered (123456..->142536..). Cosmetic changes, mostly in 1b, 2a, 3d, (4a7)
(v3 numbers) and Introductio
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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