165 research outputs found
Evasiveness and the Distribution of Prime Numbers
We confirm the eventual evasiveness of several classes of monotone graph
properties under widely accepted number theoretic hypotheses. In particular we
show that Chowla's conjecture on Dirichlet primes implies that (a) for any
graph , "forbidden subgraph " is eventually evasive and (b) all
nontrivial monotone properties of graphs with edges are
eventually evasive. ( is the number of vertices.)
While Chowla's conjecture is not known to follow from the Extended Riemann
Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's functions), we show
(b) with the bound under ERH.
We also prove unconditional results: (a) for any graph , the query
complexity of "forbidden subgraph " is ; (b) for
some constant , all nontrivial monotone properties of graphs with edges are eventually evasive.
Even these weaker, unconditional results rely on deep results from number
theory such as Vinogradov's theorem on the Goldbach conjecture.
Our technical contribution consists in connecting the topological framework
of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti,
Khot, and Shi (2002), with a deeper analysis of the orbital structure of
permutation groups and their connection to the distribution of prime numbers.
Our unconditional results include stronger versions and generalizations of some
result of Chakrabarti et al.Comment: 12 pages (conference version for STACS 2010
Collapsibility of CAT(0) spaces
Collapsibility is a combinatorial strengthening of contractibility. We relate
this property to metric geometry by proving the collapsibility of any complex
that is CAT(0) with a metric for which all vertex stars are convex. This
strengthens and generalizes a result by Crowley. Further consequences of our
work are:
(1) All CAT(0) cube complexes are collapsible.
(2) Any triangulated manifold admits a CAT(0) metric if and only if it admits
collapsible triangulations.
(3) All contractible d-manifolds () admit collapsible CAT(0)
triangulations. This discretizes a classical result by Ancel--Guilbault.Comment: 27 pages, 3 figures. The part on collapsibility of convex complexes
has been removed and forms a new paper, called "Barycentric subdivisions of
convexes complex are collapsible" (arXiv:1709.07930). The part on enumeration
of manifolds has also been removed and forms now a third paper, called "A
Cheeger-type exponential bound for the number of triangulated manifolds"
(arXiv:1710.00130
Collapsing along monotone poset maps
We introduce the notion of nonevasive reduction, and show that for any
monotone poset map , the simplicial complex {\tt
NE}-reduces to , for any .
As a corollary, we prove that for any order-preserving map
satisfying , for any , the simplicial complex
collapses to . We also obtain a generalization of
Crapo's closure theorem.Comment: To appear in the International Journal of Mathematics and
Mathematical Science
Vertex decompositions of two-dimensional complexes and graphs
We investigate families of two-dimensional simplicial complexes defined in
terms of vertex decompositions. They include nonevasive complexes, strongly
collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary
and Palmer. We investigate the complexity of recognition problems for those
families and some of their combinatorial properties. Certain results follow
from analogous decomposition techniques for graphs. For example, we prove that
it is NP-complete to decide if a graph can be reduced to a discrete graph by a
sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug
Order complexes of noncomplemented lattices are nonevasive
We reprove and generalize in a combinatorial way the result of A. Bj\"orner
[J.\ Comb.\ Th.\ A {\bf 30}, 1981, pp.~90--100, Theorem 3.3], that order
complexes of noncomplemented lattices are contractible, namely by showing that
these simplicial complexes are in fact nonevasive, in particular collapsible
One-Point Suspensions and Wreath Products of Polytopes and Spheres
It is known that the suspension of a simplicial complex can be realized with
only one additional point. Suitable iterations of this construction generate
highly symmetric simplicial complexes with various interesting combinatorial
and topological properties. In particular, infinitely many non-PL spheres as
well as contractible simplicial complexes with a vertex-transitive group of
automorphisms can be obtained in this way.Comment: 17 pages, 8 figure
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