547 research outputs found
Irreducible factors of modular representations of mapping class groups arising in Integral TQFT
We find decomposition series of length at most two for modular
representations in positive characteristic of mapping class groups of surfaces
induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at
the p-th root of unity, where p is an odd prime. The dimensions of the
irreducible factors are given by Verlinde-type formulas.Comment: 29 pages, two conjectures made in Remark 7.3 of version 1 are now
proved in the added subsection 7.5; simplified equation (5); added Remark
7.5; rewrote parts of section 4 to make paper more self-containe
Integral Lattices in TQFT
We find explicit bases for naturally defined lattices over a ring of
algebraic integers in the SO(3) TQFT-modules of surfaces at roots of unity of
odd prime order. Some applications relating quantum invariants to classical
3-manifold topology are given.Comment: 31 pages, v2: minor modifications. To appear in Ann. Sci. Ecole Norm.
Su
Notes on complexity of packing coloring
A packing -coloring for some integer of a graph is a mapping
such that any two vertices of color
are in distance at least . This concept
is motivated by frequency assignment problems. The \emph{packing chromatic
number} of is the smallest such that there exists a packing
-coloring of .
Fiala and Golovach showed that determining the packing chromatic number for
chordal graphs is \NP-complete for diameter exactly 5. While the problem is
easy to solve for diameter 2, we show \NP-completeness for any diameter at
least 3. Our reduction also shows that the packing chromatic number is hard to
approximate within for any .
In addition, we design an \FPT algorithm for interval graphs of bounded
diameter. This leads us to exploring the problem of finding a partial coloring
that maximizes the number of colored vertices.Comment: 9 pages, 2 figure
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
Cyclic operads and homology of graph complexes
We will consider P-graph complexes, where P is a cyclic operad. P-graph
complexes are natural generalizations of Kontsevich's graph complexes -- for P
= the operad for associative algebras it is the complex of ribbon graphs, for P
= the operad for commutative associative algebras, the complex of all graphs.
We construct a `universal class' in the cohomology of the graph complex with
coefficients in a theory. The Kontsevich-type invariant is then an evaluation,
on a concrete cyclic algebra, of this class. We also explain some results of M.
Penkava and A. Schwarz on the construction of an invariant from a cyclic
deformation of a cyclic algebra. Our constructions are illustrated by a `toy
model' of tree complexes.Comment: LaTeX 2.09 + article12pt,leqno style, 10 page
Finiteness conditions for graph algebras over tropical semirings
Connection matrices for graph parameters with values in a field have been
introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph
parameters with connection matrices of finite rank can be computed in
polynomial time on graph classes of bounded tree-width. We introduce join
matrices, a generalization of connection matrices, and allow graph parameters
to take values in the tropical rings (max-plus algebras) over the real numbers.
We show that rank-finiteness of join matrices implies that these graph
parameters can be computed in polynomial time on graph classes of bounded
clique-width. In the case of graph parameters with values in arbitrary
commutative semirings, this remains true for graph classes of bounded linear
clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that
definability of a graph parameter in Monadic Second Order Logic implies rank
finiteness. We also show that there are uncountably many integer valued graph
parameters with connection matrices or join matrices of fixed finite rank. This
shows that rank finiteness is a much weaker assumption than any definability
assumption.Comment: 12 pages, accepted for presentation at FPSAC 2014 (Chicago, June 29
-July 3, 2014), to appear in Discrete Mathematics and Theoretical Computer
Scienc
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