An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm

Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most important and enticing open problems in coloring, such as Reed's $\omega, \Delta, \chi$ Conjecture, follow this theme. This thesis both broadens and deepens this classical paradigm. In Part~1, we tackle list-coloring problems in which the number of colors available to each vertex $v$ depends on its degree, denoted $d(v)$, and the size of the largest clique containing it, denoted $\omega(v)$. We make extensive use of the probabilistic method in this part. We conjecture the ``list-local version'' of Reed's Conjecture, that is every graph is $L$-colorable if $L$ is a list-assignment such that $$|L(v)| \geq \lceil (1 - \varepsilon)(d(v) + 1) + \varepsilon\omega(v))\rceil$$ for each vertex $v$ and $\varepsilon \leq 1/2$, and we prove this for $\varepsilon \leq 1/330$ under some mild additional assumptions. We also conjecture the ``$\mathrm{mad}$ version'' of Reed's Conjecture, even for list-coloring. That is, for $\varepsilon \leq 1/2$, every graph $G$ satisfies \chi_\ell(G) \leq \lceil (1 - \varepsilon)(\mad(G) + 1) + \varepsilon\omega(G)\rceil, where $\mathrm{mad}(G)$ is the maximum average degree of $G$. We prove this conjecture for small values of $\varepsilon$, assuming $\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G)$. We actually prove a stronger result that improves bounds on the density of critical graphs without large cliques, a long-standing problem, answering a question of Kostochka and Yancey. In the proof, we use a novel application of the discharging method to find a set of vertices for which any precoloring can be extended to the remainder of the graph using the probabilistic method. Our result also makes progress towards Hadwiger's Conjecture: we improve the best known bound on the chromatic number of $K_t$-minor free graphs by a constant factor. We provide a unified treatment of coloring graphs with small clique number. We prove that for $\Delta$ sufficiently large, if $G$ is a graph of maximum degree at most $\Delta$ with list-assignment $L$ such that for each vertex $v\in V(G)$, $$|L(v)| \geq 72\cdot d(v)\min\left\{\sqrt{\frac{\ln(\omega(v))}{\ln(d(v))}}, \frac{\omega(v)\ln(\ln(d(v)))}{\ln(d(v))}, \frac{\log_2(\chi(G[N(v)]) + 1)}{\ln(d(v))}\right\}$$ and $d(v) \geq \ln^2\Delta$, then $G$ is $L$-colorable. This result simultaneously implies three famous results of Johansson from the 90s, as well as the following new bound on the chromatic number of any graph $G$ with $\omega(G)\leq \omega$ and $\Delta(G)\leq \Delta$ for $\Delta$ sufficiently large: $$\chi(G) \leq 72\Delta\sqrt{\frac{\ln\omega}{\ln\Delta}}.$$ In Part~2, we introduce and develop the theory of fractional coloring with local demands. A fractional coloring of a graph is an assignment of measurable subsets of the $[0, 1]$-interval to each vertex such that adjacent vertices receive disjoint sets, and we think of vertices ``demanding'' to receive a set of color of comparatively large measure. We prove and conjecture ``local demands versions'' of various well-known coloring results in the $\omega, \Delta, \chi$ paradigm, including Vizing's Theorem and Molloy's recent breakthrough bound on the chromatic number of triangle-free graphs. The highlight of this part is the ``local demands version'' of Brooks' Theorem. Namely, we prove that if $G$ is a graph and $f : V(G) \rightarrow [0, 1]$ such that every clique $K$ in $G$ satisfies $\sum_{v\in K}f(v) \leq 1$ and every vertex $v\in V(G)$ demands $f(v) \leq 1/(d(v) + 1/2)$, then $G$ has a fractional coloring $\phi$ in which the measure of $\phi(v)$ for each vertex $v\in V(G)$ is at least $f(v)$. This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle

Grained integers and applications to cryptography

To meet the requirements of the modern communication society, cryptographic techniques are of central importance. In modern cryptography, we try to build cryptographic primitives, whose security can be reduced to solving a particular number theoretic problem for which no fast algorithmic method is known by now. Thus, any advance in the understanding of the nature of such problems indirectly gives insight in the analysis of some of the most practical cryptographic techniques. In this work we analyze exactly this aspect much more deeply: How can we use some of the purely theoretical results in number theory to answer very practical questions on the security of widely used cryptographic algorithms and how can we use such results in concrete implementations? While trying to answer these kinds of security-related questions, we always think two-fold: From a cryptographic, security-ensuring perspective and from a cryptanalytic one. After we outlined -- with a special focus on the historical development of these results -- the necessary analytic and algorithmic foundations of number theory, we first delve into the question how point addition on certain elliptic curves can be done efficiently. The resulting formulas have their application in the cryptanalysis of crypto systems that are insecure if factoring integers can be done efficiently. The rest of the thesis is devoted to the study of integers, all of whose prime factors are neither too small nor too large. We show with the help of two applications how one can use the properties of such kinds of integers to answer very practical questions in the design and the analysis of cryptographic primitives: The optimization of a hardware-realization of the cofactorization step of the General Number Field Sieve and the analysis of different standardized key-generation algorithms

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

On Multilevel Methods Based on Non-Nested Meshes

This thesis is concerned with multilevel methods for the efficient solution of partial differential equations in the field of scientific computing. Further, emphasis is put on an extensive study of the information transfer between finite element spaces associated with non-nested meshes. For the discretization of complicated geometries with a finite element method, unstructured meshes are often beneficial as they can easily be adjusted to the shape of the computational domain. Such meshes, and thus the corresponding discrete function spaces, do not allow for straightforward multilevel hierarchies that could be exploited to construct fast solvers. In the present thesis, we present a class of "semi-geometric" multilevel iterations, which are based on hierarchies of independent, non-nested meshes. This is realized by a variational approach such that the images of suitable prolongation operators in the next (finer) space recursively determine the coarse level spaces. The semi-geometric concept is of very general nature compared with other methods relying on geometric considerations. This is reflected in the relatively loose relations of the employed meshes to each other. The specific benefit of the approach based on non-nested meshes is the flexibility in the choice of the coarse meshes, which can, for instance, be generated independently by standard methods. The resolution of the boundaries of the actual computational domain in the constructed coarse level spaces is a characteristic feature of the devised class of methods. The flexible applicability and the efficiency of the presented solution methods is demonstrated in a series of numerical experiments. We also explain the practical implementation of the semi-geometric ideas and concrete transfer concepts between non-nested meshes. Moreover, an extension to a semi-geometric monotone multigrid method for the solution of variational inequalities is discussed. We carry out the analysis of the convergence and preconditioning properties, respectively, in the framework of the theory of subspace correction methods. Our technical considerations yield a quasi-optimal result, which we prove for general, shape regular meshes by local arguments. The relevant properties of the operators for the prolongation between non-nested finite element spaces are the H1-stability and an L2-approximation property as well as the locality of the transfer. This thesis is a contribution to the development of fast solvers for equations on complicated geometries with focus on geometric techniques (as opposed to algebraic ones). Connections to other approaches are carefully elaborated. In addition, we examine the actual information transfer between non-nested finite element spaces. In a novel study, we combine theoretical, practical and experimental considerations. A thourough investigation of the qualitative properties and a quantitative analysis of the differences of individual transfer concepts to each other lead to new results on the information transfer as such. Finally, by the introduction of a generalized projection operator, the pseudo-L2-projection, we obtain a significantly better approximation of the actual L2-orthogonal projection than other approaches from the literature.Nicht-geschachtelte Gitter in Multilevel-Verfahren Diese Arbeit beschäftigt sich mit Multilevel-Verfahren zur effizienten Lösung von Partiellen Differentialgleichungen im Bereich des Wissenschaftlichen Rechnens. Dabei liegt ein weiterer Schwerpunkt auf der eingehenden Untersuchung des Informationsaustauschs zwischen Finite-Elemente-Räumen zu nicht-geschachtelten Gittern. Zur Diskretisierung von komplizierten Geometrien mit einer Finite-Elemente-Methode sind unstrukturierte Gitter oft von Vorteil, weil sie der Form des Rechengebiets einfacher angepasst werden können. Solche Gitter, und somit die zugehörigen diskreten Funktionenräume, besitzen im Allgemeinen keine leicht zugängliche Multilevel-Struktur, die sich zur Konstruktion schneller Löser ausnutzen ließe. In der vorliegenden Arbeit stellen wir eine Klasse "semi-geometrischer" Multilevel-Iterationen vor, die auf Hierarchien voneinander unabhängiger, nicht-geschachtelter Gitter beruhen. Dabei bestimmen in einem variationellen Ansatz rekursiv die Bilder geeigneter Prolongationsoperatoren im jeweils folgenden (feineren) Raum die Grobgitterräume. Das semi-geometrische Konzept ist sehr allgemeiner Natur verglichen mit anderen Verfahren, die auf geometrischen Überlegungen beruhen. Dies zeigt sich in der verhältnismäßig losen Beziehung der verwendeten Gitter zueinander. Der konkrete Nutzen des Ansatzes mit nicht-geschachtelten Gittern ist die Flexibilität der Wahl der Grobgitter. Diese können beispielsweise unabhängig mit Standardverfahren generiert werden. Die Auflösung des Randes des tatsächlichen Rechengebiets in den konstruierten Grobgitterräumen ist eine Eigenschaft der entwickelten Verfahrensklasse. Die flexible Einsetzbarkeit und die Effizienz der vorgestellten Lösungsverfahren zeigt sich in einer Reihe von numerischen Experimenten. Dazu geben wir Hinweise zur praktischen Umsetzung der semi-geometrischen Ideen und konkreter Transfer-Konzepte zwischen nicht-geschachtelten Gittern. Darüber hinaus wird eine Erweiterung zu einem semi-geometrischen monotonen Mehrgitterverfahren zur Lösung von Variationsungleichungen untersucht. Wir führen die Analysis der Konvergenz- bzw. Vorkonditionierungseigenschaften im Rahmen der Theorie der Teilraumkorrekturmethoden durch. Unsere technische Ausarbeitung liefert ein quasi-optimales Resultat, das wir mithilfe lokaler Argumente für allgemeine, shape-reguläre Gitterfamilien beweisen. Als relevante Eigenschaften der Operatoren zur Prolongation zwischen nicht-geschachtelten Finite-Elemente-Räumen erweisen sich die H1-Stabilität und eine L2-Approximationseigenschaft sowie die Lokalität des Transfers. Diese Arbeit ist ein Beitrag zur Entwicklung schneller Löser für Gleichungen auf komplizierten Gebieten mit Schwerpunkt auf geometrischen Techniken (im Unterschied zu algebraischen). Verbindungen zu anderen Ansätzen werden sorgfältig aufgezeigt. Daneben untersuchen wir den Informationsaustausch zwischen nicht-geschachtelten Finite-Elemente-Räumen als solchen. In einer neuartigen Studie verbinden wir theoretische, praktische und experimentelle Überlegungen. Eine sorgfältige Prüfung der qualitativen Eigenschaften sowie eine quantitative Analyse der Unterschiede verschiedener Transfer-Konzepte zueinander führen zu neuen Ergebnissen bezüglich des Informationsaustauschs selbst. Schließlich erreichen wir durch die Einführung eines verallgemeinerten Projektionsoperators, der Pseudo-L2-Projektion, eine deutlich bessere Approximation der eigentlichen L2-orthogonalen Projektion als andere Ansätze aus der Literatur