252 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
Bounds on the diameter of Cayley graphs of the symmetric group
In this paper we are concerned with the conjecture that, for any set of
generators S of the symmetric group of degree n, the word length in terms of S
of every permutation is bounded above by a polynomial of n. We prove this
conjecture for sets of generators containing a permutation fixing at least 37%
of the points.Comment: 17 pages, 6 table
Affine Symmetries of Orbit Polytopes
An orbit polytope is the convex hull of an orbit under a finite group . We develop a general theory of possible
affine symmetry groups of orbit polytopes. For every group, we define an open
and dense set of generic points such that the orbit polytopes of generic points
have conjugated affine symmetry groups. We prove that the symmetry group of a
generic orbit polytope is again if is itself the affine symmetry group
of some orbit polytope, or if is absolutely irreducible. On the other hand,
we describe some general cases where the affine symmetry group grows.
We apply our theory to representation polytopes (the convex hull of a finite
matrix group) and show that their affine symmetries can be computed effectively
from a certain character. We use this to construct counterexamples to a
conjecture of Baumeister et~al.\ on permutation polytopes [Advances in Math.
222 (2009), 431--452, Conjecture~5.4].Comment: v2: Referee comments implemented, last section updated. Numbering of
results changed only in Sections 9 and 10. v3: Some typos corrected. Final
version as published. 36 pages, 5 figures (TikZ
Most primitive groups are full automorphism groups of edge-transitive hypergraphs
We prove that, for a primitive permutation group G acting on a set of size n,
other than the alternating group, the probability that Aut(X,Y^G) = G for a
random subset Y of X, tends to 1 as n tends to infinity. So the property of the
title holds for all primitive groups except the alternating groups and finitely
many others. This answers a question of M. Klin. Moreover, we give an upper
bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph.
This is essentially best possible.Comment: To appear in special issue of Journal of Algebra in memory of Akos
Seres
Automorphisms of Cayley graphs on generalised dicyclic groups
A graph is called a GRR if its automorphism group acts regularly on its
vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that
there are only two infinite families of finite groups that do not admit GRRs :
abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on
such a group admits specific additional graph automorphisms that depend only on
the group. Recently, Dobson and the last two authors showed that almost all
Cayley graphs on abelian groups admit no automorphisms other than these obvious
necessary ones. In this paper, we prove the analogous result for Cayley graphs
on the remaining family of exceptional groups: generalised dicyclic groups.Comment: 18 page
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