5,028 research outputs found
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Disproof of the List Hadwiger Conjecture
The List Hadwiger Conjecture asserts that every -minor-free graph is
-choosable. We disprove this conjecture by constructing a
-minor-free graph that is not -choosable for every integer
Estimating the Number of Stable Configurations for the Generalized Thomson Problem
Given a natural number N, one may ask what configuration of N points on the
two-sphere minimizes the discrete generalized Coulomb energy. If one applies a
gradient-based numerical optimization to this problem, one encounters many
configurations that are stable but not globally minimal. This led the authors
of this manuscript to the question, how many stable configurations are there?
In this manuscript we report methods for identifying and counting observed
stable configurations, and estimating the actual number of stable
configurations. These estimates indicate that for N approaching two hundred,
there are at least tens of thousands of stable configurations.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10955-015-1245-
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