566 research outputs found
Hedetniemi's conjecture for Kneser hypergraphs
One of the most famous conjecture in graph theory is Hedetniemi's conjecture
stating that the chromatic number of the categorical product of graphs is the
minimum of their chromatic numbers. Using a suitable extension of the
definition of the categorical product, Zhu proposed in 1992 a similar
conjecture for hypergraphs. We prove that Zhu's conjecture is true for the
usual Kneser hypergraphs of same rank. It provides to the best of our knowledge
the first non-trivial and explicit family of hypergraphs with rank larger than
two satisfying this conjecture (the rank two case being Hedetniemi's
conjecture). We actually prove a more general result providing a lower bound on
the chromatic number of the categorical product of any Kneser hypergraphs as
soon as they all have same rank. We derive from it new families of graphs
satisfying Hedetniemi's conjecture. The proof of the lower bound relies on the
-Tucker lemma
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
- …