39,948 research outputs found
Hendry’s Conjecture on Chordal Graph Subclasses
A cycle is extendable if there exists another cycle on the same set of vertices plus one more vertex. G.R.T. Hendry conjectured (1990) that every non spanning cycle in a Hamiltonian chordal graph is extendable. This has recently been disproved (2015), but is still open for classes of strongly chordal graphs. Hendry’s Conjecture has been shown to hold for the following subclasses of chordal graphs: planar chordal graphs (2002), interval graphs, strongly chordal graphs with (two specific) forbidden subgraphs, split graphs (2006), and spider intersection graphs (2013). Chapter 1 of this dissertation is an introduction to the subject matter. In chapter 2 we verify that Hendry’s Conjecture holds for Ptolemaic graphs which are a subclass of strongly chordal graphs, alongside with a strong result on how smoothly the extension can happen. In chapter 3 we develop tools for working on tree representations of chordal graphs with Hendry’s Conjecture in mind. Chapter 4 is an application of these tools to interval graphs, another subclass of chordal graphs. Chapter 5 is about manipulating the aformentioned counterexample to Hendry’s Conjecture, and applying tools from chapter 3 on it. This yields information on the structure of graphs for which Hendry’s conjecture holds
Generalized Gr\"otzsch Graphs
The aim of this paper is to present a generalization of Gr\"otzsch graph.
Inspired by structure of the Gr\"otzsch's graph, we present constructions of
two families of graphs, and for odd and even values of
respectively and on vertices. We show that each member of this
family is non-planar, triangle-free, and Hamiltonian. Further, when is odd
the graph is maximal triangle-free, and when is even, the addition of
exactly edges makes the graph maximal triangle-free. We
show that is 4-chromatic and is 3-chromatic for all . Further,
we note some other properties of these graphs and compare with Mycielski's
construction.Comment: This is a first draft report about ongoing work on the Gr\"otzsch
Graph
Foliations of Isonergy Surfaces and Singularities of Curves
It is well known that changes in the Liouville foliations of the isoenergy
surfaces of an integrable system imply that the bifurcation set has
singularities at the corresponding energy level. We formulate certain
genericity assumptions for two degrees of freedom integrable systems and we
prove the opposite statement: the essential critical points of the bifurcation
set appear only if the Liouville foliations of the isoenergy surfaces change at
the corresponding energy levels. Along the proof, we give full classification
of the structure of the isoenergy surfaces near the critical set under our
genericity assumptions and we give their complete list using Fomenko graphs.
This may be viewed as a step towards completing the Smale program for relating
the energy surfaces foliation structure to singularities of the momentum
mappings for non-degenerate integrable two degrees of freedom systems.Comment: 30 pages, 19 figure
On the S-matrix renormalization in effective theories
This is the 5-th paper in the series devoted to explicit formulating of the
rules needed to manage an effective field theory of strong interactions in
S-matrix sector. We discuss the principles of constructing the meaningful
perturbation series and formulate two basic ones: uniformity and summability.
Relying on these principles one obtains the bootstrap conditions which restrict
the allowed values of the physical (observable) parameters appearing in the
extended perturbation scheme built for a given localizable effective theory.
The renormalization prescriptions needed to fix the finite parts of
counterterms in such a scheme can be divided into two subsets: minimal --
needed to fix the S-matrix, and non-minimal -- for eventual calculation of
Green functions; in this paper we consider only the minimal one. In particular,
it is shown that in theories with the amplitudes which asymptotic behavior is
governed by known Regge intercepts, the system of independent renormalization
conditions only contains those fixing the counterterm vertices with
lines, while other prescriptions are determined by self-consistency
requirements. Moreover, the prescriptions for cannot be taken
arbitrary: an infinite number of bootstrap conditions should be respected. The
concept of localizability, introduced and explained in this article, is closely
connected with the notion of resonance in the framework of perturbative QFT. We
discuss this point and, finally, compare the corner stones of our approach with
the philosophy known as ``analytic S-matrix''.Comment: 28 pages, 10 Postscript figures, REVTeX4, submitted to Phys. Rev.
The resultant parameters of effective theory
This is the 4-th paper in the series devoted to a systematic study of the
problem of mathematically correct formulation of the rules needed to manage an
effective field theory. Here we consider the problem of constructing the full
set of essential parameters in the case of the most general effective
scattering theory containing no massless particles with spin J > 1/2. We
perform the detailed classification of combinations of the Hamiltonian coupling
constants and select those which appear in the expressions for renormalized
S-matrix elements at a given loop order.Comment: 21 pages, 4 LaTeX figures, submitted to Phys. Rev.
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