53,144 research outputs found
On the Typical Structure of Graphs in a Monotone Property
Given a graph property , it is interesting to determine the
typical structure of graphs that satisfy . In this paper, we
consider monotone properties, that is, properties that are closed under taking
subgraphs. Using results from the theory of graph limits, we show that if
is a monotone property and is the largest integer for which
every -colorable graph satisfies , then almost every graph with
is close to being a balanced -partite graph.Comment: 5 page
Singular curves and quasi-hereditary algebras
In this article we construct a categorical resolution of singularities of an
excellent reduced curve , introducing a certain sheaf of orders on . This
categorical resolution is shown to be a recollement of the derived category of
coherent sheaves on the normalization of and the derived category of finite
length modules over a certain artinian quasi-hereditary ring depending
purely on the local singularity types of .
Using this technique, we prove several statements on the Rouquier dimension
of the derived category of coherent sheaves on . Moreover, in the case
is rational and projective we construct a finite dimensional quasi-hereditary
algebra such that the triangulated category of perfect complexes on
embeds into as a full subcategory.Comment: minor changes; to appear in IMR
New results on word-representable graphs
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
for each . The set of word-representable graphs
generalizes several important and well-studied graph families, such as circle
graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at
most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and
A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202.
Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in
Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.],
in the present paper we show that not all graphs of vertex degree at most 4 are
word-representable. Combining this result with some previously known facts, we
derive that the number of -vertex word-representable graphs is
Hereditary properties of combinatorial structures: posets and oriented graphs
A hereditary property of combinatorial structures is a collection of
structures (e.g. graphs, posets) which is closed under isomorphism, closed
under taking induced substructures (e.g. induced subgraphs), and contains
arbitrarily large structures. Given a property P, we write P_n for the
collection of distinct (i.e., non-isomorphic) structures in a property P with n
vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of
P. Also, we write P^n for the collection of distinct labelled structures in P
with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled
speed of P.
The possible labelled speeds of a hereditary property of graphs have been
extensively studied, and the aim of this paper is to investigate the possible
speeds of other combinatorial structures, namely posets and oriented graphs.
More precisely, we show that (for sufficiently large n), the labelled speed of
a hereditary property of posets is either 1, or exactly a polynomial, or at
least 2^n - 1. We also show that there is an initial jump in the possible
unlabelled speeds of hereditary properties of posets, tournaments and directed
graphs, from bounded to linear speed, and give a sharp lower bound on the
possible linear speeds in each case.Comment: 26 pgs, no figure
1-quasi-hereditary algebras
Motivated by the structure of the algebras associated to the blocks of the
BGG-category O we define a subclass of quasi-hereditary algebras called
1-quasi-hereditary. Many properties of these algebras only depend on the
defining partial order. In particular, we can determine the quiver and the form
of the relations. Moreover, if the Ringel dual of a 1-quasi-hereditary algebra
is also 1-quasi-hereditary, then the structure of the characteristic tilting
module can be computed.Comment: 20 pages, examples and some statements are removed and will appear in
a separate file. Some proofs are rewritte
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