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    Hendry’s Conjecture on Chordal Graph Subclasses

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    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

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    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, GmG_m and HmH_m for odd and even values of mm respectively and on n=2m+1n = 2m +1 vertices. We show that each member of this family is non-planar, triangle-free, and Hamiltonian. Further, when mm is odd the graph GmG_m is maximal triangle-free, and when mm is even, the addition of exactly m2\frac{m}{2} edges makes the graph HmH_m maximal triangle-free. We show that GmG_m is 4-chromatic and HmH_m is 3-chromatic for all mm. 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

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    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

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    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 n3n \leq 3 lines, while other prescriptions are determined by self-consistency requirements. Moreover, the prescriptions for n3n \leq 3 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

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    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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