1,188 research outputs found
Buchstaber numbers and classical invariants of simplicial complexes
Buchstaber invariant is a numerical characteristic of a simplicial complex,
arising from torus actions on moment-angle complexes. In the paper we study the
relation between Buchstaber invariants and classical invariants of simplicial
complexes such as bigraded Betti numbers and chromatic invariants. The
following two statements are proved. (1) There exists a simplicial complex U
with different real and ordinary Buchstaber invariants. (2) There exist two
simplicial complexes with equal bigraded Betti numbers and chromatic numbers,
but different Buchstaber invariants. To prove the first theorem we define
Buchstaber number as a generalized chromatic invariant. This approach allows to
guess the required example. The task then reduces to a finite enumeration of
possibilities which was done using GAP computational system. To prove the
second statement we use properties of Taylor resolutions of face rings.Comment: 19 pages, 2 figure
The Coloring Ideal and Coloring Complex of a Graph
Let be a simple graph on vertices. We define a monomial ideal in
the Stanley-Reisner ring of the order complex of the Boolean algebra on
atoms. The monomials in are in one-to-one correspondence with the proper
colorings of . In particular, the Hilbert polynomial of equals the
chromatic polynomial of .
The ideal is generated by square-free monomials, so is the
Stanley-Reisner ring of a simplicial complex . The -vector of is a
certain transformation of the tail of the chromatic polynomial
of . The combinatorial structure of the complex is described
explicitly and it is shown that the Euler characteristic of equals the
number of acyclic orientations of .Comment: 13 pages, 3 figure
Box complexes, neighborhood complexes, and the chromatic number
Lovasz's striking proof of Kneser's conjecture from 1978 using the
Borsuk--Ulam theorem provides a lower bound on the chromatic number of a graph.
We introduce the shore subdivision of simplicial complexes and use it to show
an upper bound to this topological lower bound and to construct a strong
Z_2-deformation retraction from the box complex (in the version introduced by
Matousek and Ziegler) to the Lovasz complex. In the process, we analyze and
clarify the combinatorics of the complexes involved and link their structure
via several ``intermediate'' complexes.Comment: 8 pages, 1 figur
Topological lower bounds for the chromatic number: A hierarchy
This paper is a study of ``topological'' lower bounds for the chromatic
number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978,
in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology.
This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with
all -element subsets of as vertices and all pairs of
disjoint sets as edges, has chromatic number . Several other proofs
have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz,
Greene, and others), all of them based on some version of the Borsuk--Ulam
theorem, but otherwise quite different. Each can be extended to yield some
lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe
that \emph{every} finite graph may be represented as a generalized Kneser
graph, to which the above bounds apply.)
We show that these bounds are almost linearly ordered by strength, the
strongest one being essentially Lov\'asz' original bound in terms of a
neighborhood complex. We also present and compare various definitions of a
\emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz
and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but
the construction is simpler and functorial, mapping graphs with homomorphisms
to -spaces with -maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea
Colouring quadrangulations of projective spaces
A graph embedded in a surface with all faces of size 4 is known as a
quadrangulation. We extend the definition of quadrangulation to higher
dimensions, and prove that any graph G which embeds as a quadrangulation in the
real projective space P^n has chromatic number n+2 or higher, unless G is
bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996),
219-227]. The family of quadrangulations of projective spaces includes all
complete graphs, all Mycielski graphs, and certain graphs homomorphic to
Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser
theorem
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