30,413 research outputs found
Tverberg's theorem and graph coloring
The topological Tverberg theorem has been generalized in several directions
by setting extra restrictions on the Tverberg partitions.
Restricted Tverberg partitions, defined by the idea that certain points
cannot be in the same part, are encoded with graphs. When two points are
adjacent in the graph, they are not in the same part. If the restrictions are
too harsh, then the topological Tverberg theorem fails. The colored Tverberg
theorem corresponds to graphs constructed as disjoint unions of small complete
graphs. Hell studied the case of paths and cycles.
In graph theory these partitions are usually viewed as graph colorings. As
explored by Aharoni, Haxell, Meshulam and others there are fundamental
connections between several notions of graph colorings and topological
combinatorics.
For ordinary graph colorings it is enough to require that the number of
colors q satisfy q>Delta, where Delta is the maximal degree of the graph. It
was proven by the first author using equivariant topology that if q>\Delta^2
then the topological Tverberg theorem still works. It is conjectured that
q>K\Delta is also enough for some constant K, and in this paper we prove a
fixed-parameter version of that conjecture.
The required topological connectivity results are proven with shellability,
which also strengthens some previous partial results where the topological
connectivity was proven with the nerve lemma.Comment: To appear in Discrete and Computational Geometry, 13 pages, 1 figure.
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On Topological Indices And Domination Numbers Of Graphs
Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index (PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well
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
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