208 research outputs found
Constructive Algebraic Topology
The classical ``computation'' methods in Algebraic Topology most often work
by means of highly infinite objects and in fact +are_not+ constructive. Typical
examples are shown to describe the nature of the problem. The Rubio-Sergeraert
solution for Constructive Algebraic Topology is recalled. This is not only a
theoretical solution: the concrete computer program +Kenzo+ has been written
down which precisely follows this method. This program has been used in various
cases, opening new research subjects and producing in several cases significant
results unreachable by hand. In particular the Kenzo program can compute the
first homotopy groups of a simply connected +arbitrary+ simplicial set.Comment: 24 pages, background paper for a plenary talk at the EACA Congress of
Tenerife, September 199
-Critical Graphs in -Free Graphs
Given two graphs and , a graph is -free if it
contains no induced subgraph isomorphic to or . Let be the
path on vertices. A graph is -vertex-critical if has chromatic
number but every proper induced subgraph of has chromatic number less
than . The study of -vertex-critical graphs for graph classes is an
important topic in algorithmic graph theory because if the number of such
graphs that are in a given hereditary graph class is finite, then there is a
polynomial-time algorithm to decide if a graph in the class is
-colorable.
In this paper, we initiate a systematic study of the finiteness of
-vertex-critical graphs in subclasses of -free graphs. Our main result
is a complete classification of the finiteness of -vertex-critical graphs in
the class of -free graphs for all graphs on 4 vertices. To obtain
the complete dichotomy, we prove the finiteness for four new graphs using
various techniques -- such as Ramsey-type arguments and the dual of Dilworth's
Theorem -- that may be of independent interest.Comment: 18 page
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
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