19 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
Improved hardness for H-colourings of G-colourable graphs
We present new results on approximate colourings of graphs and, more
generally, approximate H-colourings and promise constraint satisfaction
problems.
First, we show NP-hardness of colouring -colourable graphs with
colours for every . This improves
the result of Bul\'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness
of colouring -colourable graphs with colours for , and the
result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring
-colourable graphs with colours for sufficiently large .
Thus, for , we improve from known linear/sub-exponential gaps to
exponential gaps.
Second, we show that the topology of the box complex of H alone determines
whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite,
H-colourable) G. This formalises the topological intuition behind the result of
Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is
NP-hard for all (3-colourable, non-bipartite) G. We use this technique to
establish NP-hardness of H-colouring of G-colourable graphs for H that include
but go beyond , including square-free graphs and circular cliques (leaving
and larger cliques open).
Underlying all of our proofs is a very general observation that adjoint
functors give reductions between promise constraint satisfaction problems.Comment: Mention improvement in Proposition 2.5. SODA 202