356 research outputs found
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an
ordered proper vertex colouring of the complex. This ideal is a monomial ideal
which is generally not squarefree. We show that such a monomial ideal has a
linear resolution, as do all of its powers, if and only if the colouring
satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular
resolution supported on a cubical complex. From this, we give the graded Betti
numbers in terms of the face-vector of the underlying simplicial complex.
Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both
the ideal and its quotient. We also give explicit formul\ae\ for the
codimension, Krull dimension, multiplicity, projective dimension, depth, and
regularity. Further still, we describe the associated primes, and we show that
they are persistent.Comment: 34 pages, 8 figure
An Abstraction of Whitney's Broken Circuit Theorem
We establish a broad generalization of Whitney's broken circuit theorem on
the chromatic polynomial of a graph to sums of type
where is a finite set and is a mapping from the power set of into
an abelian group. We give applications to the domination polynomial and the
subgraph component polynomial of a graph, the chromatic polynomial of a
hypergraph, the characteristic polynomial and Crapo's beta invariant of a
matroid, and the principle of inclusion-exclusion. Thus, we discover several
known and new results in a concise and unified way. As further applications of
our main result, we derive a new generalization of the maximums-minimums
identity and of a theorem due to Blass and Sagan on the M\"obius function of a
finite lattice, which generalizes Rota's crosscut theorem. For the classical
M\"obius function, both Euler's totient function and its Dirichlet inverse, and
the reciprocal of the Riemann zeta function we obtain new expansions involving
the greatest common divisor resp. least common multiple. We finally establish
an even broader generalization of Whitney's broken circuit theorem in the
context of convex geometries (antimatroids).Comment: 18 page
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
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
Spectra of Uniform Hypergraphs
We present a spectral theory of hypergraphs that closely parallels Spectral
Graph Theory. A number of recent developments building upon classical work has
led to a rich understanding of "hyperdeterminants" of hypermatrices, a.k.a.
multidimensional arrays. Hyperdeterminants share many properties with
determinants, but the context of multilinear algebra is substantially more
complicated than the linear algebra required to address Spectral Graph Theory
(i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of
a hypermatrix via its characteristic polynomial as well as variationally. We
apply this notion to the "adjacency hypermatrix" of a uniform hypergraph, and
prove a number of natural analogues of basic results in Spectral Graph Theory.
Open problems abound, and we present a number of directions for further study.Comment: 32 pages, no figure
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