1,269 research outputs found
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
Neighborhood complexes and Kronecker double coverings
The neighborhood complex is a simplicial complex assigned to a graph
whose connectivity gives a lower bound for the chromatic number of . We
show that if the Kronecker double coverings of graphs are isomorphic, then
their neighborhood complexes are isomorphic. As an application, for integers
and greater than 2, we construct connected graphs and such that
but and . We also construct a
graph such that and the Kneser graph are not
isomorphic but their Kronecker double coverings are isomorphic.Comment: 10 pages. Some results concerning box complexes are deleted. to
appear in Osaka J. Mat
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
Generalization of neighborhood complexes
We introduce the notion of r-neighborhood complex for a positive integer r,
which is a natural generalization of Lovasz neighborhood complex. The
topologies of these complexes give some obstructions of the existence of graph
maps. We applied these complexes to prove the nonexistence of graph maps about
Kneser graphs. We prove that the fundamental groups of r-neighborhood complexes
are closely related to the (2r)-fundamental groups defined in the author's
previous paper.Comment: 8 page
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