A brief history of edge-colorings – with personal reminiscences

In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Dénes König to very recent wor

Graph theory in America 1876-1950

This narrative is a history of the contributions made to graph theory in the United States of America by American mathematicians and others who supported the growth of scholarship in that country, between the years 1876 and 1950. The beginning of this period coincided with the opening of the first research university in the United States of America, The Johns Hopkins University (although undergraduates were also taught), providing the facilities and impetus for the development of new ideas. The hiring, from England, of one of the foremost mathematicians of the time provided the necessary motivation for research and development for a new generation of American scholars. In addition, it was at this time that home-grown research mathematicians were first coming to prominence. At the beginning of the twentieth century European interest in graph theory, and to some extent the four-colour problem, began to wane. Over three decades, American mathematicians took up this field of study - notably, Oswald Veblen, George Birkhoff, Philip Franklin, and Hassler Whitney. It is necessary to stress that these four mathematicians and all the other scholars mentioned in this history were not just graph theorists but worked in many other disciplines. Indeed, they not only made significant contributions to diverse fields but, in some cases, they created those fields themselves and set the standards for others to follow. Moreover, whilst they made considerable contributions to graph theory in general, two of them developed important ideas in connection with the four-colour problem. Grounded in a paper by Alfred Bray Kempe that was notorious for its fallacious 'proof' of the four-colour theorem, these ideas were the concepts of an unavoidable set and a reducible configuration. To place the story of these scholars within the history of mathematics, America, and graph theory, brief accounts are presented of the early years of graph theory, the early years of mathematics and graph theory in the USA, and the effects of the founding of the first institute for postgraduate study in America. Additionally, information has been included on other influences by such global events as the two world wars, the depression, the influx of European scholars into the United States of America, mainly during the 1930s, and the parallel development of graph theory in Europe. Until the end of the nineteenth century, graph theory had been almost entirely the prerogative of European mathematicians. Perhaps the first work in graph theory carried out in America was by Charles Sanders Peirce, arguably America's greatest logician and philosopher at the time. In the 1860s, he studied the four-colour conjecture and claimed to have written at least two papers on the subject during that decade, but unfortunately neither of these has survived. William Edward Story entered the field in 1879, with unfortunate consequences, but it was not until 1897 that an American mathematician presented a lecture on the subject, albeit only to have the paper disappear. Paul Wernicke presented a lecture on the four-colour problem to the American Mathematician Society, but again the paper has not survived. However, his 1904 paper has survived and added to the story of graph theory, and particularly the four-colour conjecture. The year 1912 saw the real beginning of American graph theory with Veblen and Birkhoff publishing major contributions to the subject. It was around this time that European mathematicians appeared to lose interest in graph theory. In the period 1912 to 1950 much of the progress made in the subject was from America and by 1950 not only had the United States of America become the foremost country for mathematics, it was the leading centre for graph theory

Desenvolvimentos da Conjetura de Fulkerson

Orientador: Christiane Neme CamposDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Em 1971, Fulkerson propôs a seguinte conjetura: todo grafo cúbico sem arestas de corte admite seis emparelhamentos perfeitos tais que cada aresta do grafo pertence a exatamente dois destes emparelhamentos. A Conjetura de Fulkerson tem desafiado pesquisadores desde sua publicação. Esta conjetura é facilmente verificada para grafos cúbicos 3-aresta-coloráveis. Portanto, a dificuldade do problema reside em estabelecer a conjetura para grafos cúbicos sem arestas de corte que não possuem 3-coloração de arestas. Estes grafos são chamados snarks. Nesta dissertação, a Conjetura de Fulkerson e os snarks são introduzidos com ¿ênfase em sua história e resultados mais relevantes. Alguns resultados relacionados à Conjetura de Fulkerson são apresentados, enfatizando suas conexões com outras conjeturas. Um breve histórico do Problema das Quatro Cores e suas relações com snarks também são apresentados. Na segunda parte deste trabalho, a Conjetura de Fulkerson é verificada para algumas famílias infinitas de snarks construídas com o método de Loupekine, utilizando subgrafos do Grafo de Petersen. Primeiramente, mostramos que a família dos LP0-snarks satisfaz a Conjetura de Fulkerson. Em seguida, generalizamos este resultado para a família mais abrangente dos LP1-snarks. Além disto, estendemos estes resultados para Snarks de Loupekine construídos com subgrafos de snarks diferentes do Grafo de PetersenAbstract: In 1971, Fulkerson proposed a conjecture that states that every bridgeless cubic graph has six perfect matchings such that each edge of the graph belongs to precisely two of these matchings. Fulkerson's Conjecture has been challenging researchers since its publication. It is easily verified for 3-edge-colourable cubic graphs. Therefore, the difficult task is to settle the conjecture for non-3-edge-colourable bridgeless cubic graphs, called snarks. In this dissertation, Fulkerson's Conjecture and snarks are presented with emphasis in their history and remarkable results. We selected some results related to Fulkerson's Conjecture, emphasizing their reach and connections with other conjectures. It is also presented a brief history of the Four-Colour Problem and its connections with snarks. In the second part of this work, we verify Fulkerson's Conjecture for some infinite families of snarks constructed with Loupekine's method using subgraphs of the Petersen Graph. More specifically, we first show that the family of LP0-snarks satisfies Fulkerson's Conjecture. Then, we generalise this result by proving that Fulkerson's Conjecture holds for the broader family of LP1-snarks. We also extend these results to even more general Loupekine Snarks constructed with subgraphs of snarks other than the Petersen GraphMestradoCiência da ComputaçãoMestre em Ciência da Computaçã

Evaluating different management strategies to increase the effectiveness of winter cover crops as an integrated weed management measure

Weed control in agricultural production systems is indispensable to achieve stable crop yields. Integrated cropping systems are demanding for preventive and ecologically harmless weed control measures in order to protect soil and water resources and to retard the selection of herbicide-resistant weeds. Well-established winter cover crops provide nutrient retention and soil protection and may effectively suppress weeds. This contributes to reduce chemical and mechanical fall- and spring-applied weed control practices. However, producers are cautious towards integrating cover crops in crop rotations, as their performance is related to environmental conditions and varies, therefore, significantly from season to season. To increase their integration into cropping systems, reliability on weed control by cover crops needs to improve. In the current study, management strategies such as i) the cover crop sowing method, ii) the selection of water deficit tolerating cover crop species, iii) cover crop species combinations, iv) the adjustment of the mulching date and v) tillage practices after cover crop cultivation were considered as possibilities to improve the effectiveness of cover crops to control weeds during cultivation and in the subsequent cash crop. Within the first and the second publication, the general weed and A. myosuroides control ability of a cover crops mixture during and after cultivation were compared in the field with various fall-applied tillage methods and glyphosate treatments. Due to the development of highly competitive cover crop stands, weeds were suppressed by 98% and A. myosuroides by 100% during cultivation. Therefore, cover crops were more efficient compared to glyphosate application(s), non-inversion and inversion tillage and revealed a great potential to reduce or even replace chemical and mechanical fall-applied weed control measures. The efficient A. myosuroides control during the cover crop cultivation remained until spring barley harvest. This quantifies cover crops to complement herbicide resistance management strategies. In contrast, due to the weak cover crop performance during fall-to-winter within another two experiments included in the second article, weed suppressive effects of cover crops disappeared after the cultivation of cover crops. This might have been the reason why reduced tillage and adjusted mulching dates in spring failed in contributing to expand weed suppressive effects of cover crops in these experiments. Cover crop mixtures are attributed to show a greater resilience against unfavorable conditions than pure cover crop stands which is expected to result in an increased weed suppression ability. Within article three, the weed control efficacy of pure cover crop stands was compared with species mixtures. Pure stands of Avena strigosa Schreb. and Raphanus sativus var. oleiformis Pers. provided the most efficient weed control with 83% and 72%, respectively. Cover crop species mixtures showed a weaker weed suppression ability than the most efficient pure stand. In order to improve the weed control ability of cover crop mixtures, it was evaluated that the species selection is more relevant than the species diversity. Thereby, environmental requirements, such as water and temperature demand, and weed suppression mechanisms should be considered. Weed suppression of mixtures was improved by increasing the proportions of A. strigosa and R. sativus var. oleiformis, as they were showing a susceptibility for dry conditions and combine a strong competition for resources and allelopathic interference with weeds. Within the fourth article, it was explored whether a low susceptibility of single cover crop species to water-limitations accompanies an improved weed suppression ability. A. strigosa and Sinapis alba L. showed differing suitabilities to cope with water-deficit in the greenhouse. A relation between weed suppression and water demand of cover crops at the field was not identified. Although the weed control ability of cover crops is generally narrowed under water-limited conditions, the weed suppression potential of individual species seems to be independent of their water supply. The adjustment of the cover crop sowing method, the consideration of species-specific requirements and the mixing strategies, were evaluated as being important to improve the resilience of cover crops against severe environmental conditions and their weed control ability. Investigations of cover crop mixtures with respect to single component species, their mixing ratios and seed densities, might further increase the absolute and average effectiveness of cover crops as an integrated weed management practice.Unkrautkontrolle in landwirtschaftlichen Produktionssystemen ist unerlässlich, um stabile Erträge zu erzielen. Integrierte Anbausysteme zielen darauf ab, verstärkt präventive und ökologisch unbedenkliche Unkrautkontrollmaßnahmen einzusetzen, um Boden- und Wasserressourcen zu schützen und die Selektion herbizidresistenter Unkräuter zu verzögern. Gut etablierte Winterzwischenfruchtbestände sorgen für einen Nährstoffrückhalt und schützen den Boden vor Erosion. Eine effiziente Unkrautunterdrückung durch Zwischenfrüchte kann den Einsatz von chemischen und mechanischen Stoppelbearbeitungsmaßnahmen reduzieren. Winterzwischenfrüchte sind allerdings bisher noch kein fester Bestandteil in Fruchtfolgen, da deren Entwicklung, von Jahr zu Jahr, stark variieren kann. Dadurch schwankt auch die Zuverlässigkeit der Unkrautunterdrückung. Kann diese dauerhaft gewährleistet werden, könnte der Anbau von Zwischenfrüchten zunehmend interessanter werden. In dieser Studie wurden Bewirtschaftungsstrategien wie i) die Aussaatmethode von Zwischenfrüchten, ii) die Berücksichtigung trockentoleranter Zwischenfrüchte, iii) Zwischenfruchtmischungen, iv) unterschiedliche Mulchtermine und v) die Bodenbearbeitung nach dem Zwischenfruchtanbau als Möglichkeiten zur Verbesserung der Unkrautkontrolle durch Zwischenfrüchte evaluiert. In der ersten und zweiten Veröffentlichung wurde die Unterdrückung von Unkräutern und A. myosuroides durch Zwischenfrüchte, im Vergleich zu verschiedenen im Herbst durchgeführten Bodenbearbeitungsvarianten und Glyphosatbehandlungen, beurteilt. Durch die Etablierung von konkurrenzfähigen Zwischenfruchtbeständen konnten Unkräuter und A. myosuroides während der Zwischenfruchtsaison um 98% bzw. 100% reduziert werden. Behandlungen, bei denen Glyphosat appliziert oder (wende und nicht-wendende) Bodenbearbeitung durchgeführt worden war, wiesen eine schlechtere Unkrautkontrolle auf, als Behandlungen mit Zwischenfrüchten. Der Einfluss des Zwischenfruchtanbaus auf A. myosuroides war auch noch während des Anbaus der Sommergerste erheblich. Dies bestätigt die Annahme, dass der Zwischenfruchtanbau als Maßnahme im Herbizidresistenzmanagement eingesetzt werden kann. Im Gegensatz dazu, konnten die Zwischenfrüchte, die in zwei weiteren Versuchen verwendet wurden, keine unkrautunterdrückende Wirkung in der Sommerung erzielen. Reduzierte Bodenbearbeitung und angepasste Mulchtermine im Frühjahr konnten ebenfalls nicht dazu beigetragen, die unkrautunterdrückende Wirkung des Zwischenfruchtmulchs zu verbessern. Im dritten Artikel wurde überprüft, ob Zwischenfruchtmischungen, im Vergleich zu Zwischenfruchtreinsaaten, durch ihre höhere Widerstandsfähigkeit gegenüber ungünstigen Witterungsbedingungen, eine effizientere Unkrautunterdrückung aufweisen. Reinbestände von Avena strigosa Schreb. und Raphanus sativus var. oleiformis Pers. erzielten mit 83% bzw. 72% die effizienteste Unkrautunterdrückung. Mischungen zeigten eine schwächere Unkrautunter-drückung als der effizienteste Reinbestand. Um das Unkrautunterdrückungspotential von Mischungen zu verbessern, wurde evaluiert, dass die Berücksichtigung der Artenauswahl bedeutender ist, als die Artenvielfalt. Mit zunehmenden Anteilen von A. strigosa und R. sativus var. oleiformis in der Mischung, stieg die unkrautunterdrückende Wirkung. Beide Arten zeigten eine Toleranz gegenüber Trockenheit und unterdrücken Unkräuter durch physikalische und chemische Mechanismen. Im vierten Artikel wurde untersucht, ob eine geringe Sensibilität ausgewählter Zwischenfruchtarten gegenüber Wassermangel mit einer verbesserten Unkrautunterdrückung bei Trockenheit einhergeht. Dabei wurden unterschiedliche Sensibilitäten für A. strigosa und Sinapis alba L. im Gewächshaus ermittelt. Obwohl die Unkrautkontrollfähigkeit von Zwischenfrüchten unter wasserlimitierten Bedingungen generell eingeschränkt ist, korrelierte das Unkrautunterdrückungspotenzial einzelner Arten im Feldversuch nicht mit der Wasserverfügbarkeit. Die Anpassung der Zwischenfruchtaussaat, die Berücksichtigung art-spezifischer Anforderungen und die Herangehensweise Zwischenfruchtarten zu kombinieren, stellen potentielle Ansatzpunkte dar, um die Widerstandsfähigkeit von Zwischenfrüchten gegenüber ungünstigen Bedingungen zu verbessern. Damit einhergehend, kann das Unkrautunter-drückungspotential von Zwischenfrüchten gesteigert werden. Um die absolute sowie durchschnittliche Wirksamkeit von Zwischenfrüchten als integrierte Unkrautkontroll-maßnahme zu steigern, sollten Zwischenfruchtmischungen, unter Beachtung von Einzelkomponenten sowie deren Mischungsverhältnissen und Saatstärken, weiter untersucht werden

Analysis & Synthesis of Distributed Control Systems with Sparse Interconnection Topologies

This dissertation is about control, identification, and analysis of systems with sparse interconnection topologies. We address two main research objectives relating to sparsity in control systems and networks. The first problem is optimal sparse controller synthesis, and the second one is the identification of sparse network. The first part of this dissertation starts with the chapter focusing on developing theoretical frameworks for the synthesis of optimal sparse output feedback controllers under pre-specified structural constraints. This is achieved by establishing a balance between the stability of the controller and the systems quadratic performance. Our approach is mainly based on converting the problem into rank constrained optimizations.We then propose a new approach in the syntheses of sparse controllers by em- ploying the concept of Hp approximations. Considering the trade-off between the controller sparsity and the performance deterioration due to the sparsification pro- cess, we propose solving methodologies in order to obtain robust sparse controllers when the system is subject to parametric uncertainties.Next, we pivot our attention to a less-studied notion of sparsity, namely row sparsity, in our optimal controller design. Combining the concepts from the majorization theory and our proposed rank constrained formulation, we propose an exact reformulation of the optimal state feedback controllers with strict row sparsity constraint, which can be sub-optimally solved by our proposed iterative optimization techniques. The second part of this dissertation focuses on developing a theoretical framework and algorithms to derive linear ordinary differential equation models of gene regulatory networks using literature curated data and micro-array data. We propose several algorithms to derive stable sparse network matrices. A thorough comparison of our algorithms with the existing methods are also presented by applying them to both synthetic and experimental data-sets

AI: Limits and Prospects of Artificial Intelligence

The emergence of artificial intelligence has triggered enthusiasm and promise of boundless opportunities as much as uncertainty about its limits. The contributions to this volume explore the limits of AI, describe the necessary conditions for its functionality, reveal its attendant technical and social problems, and present some existing and potential solutions. At the same time, the contributors highlight the societal and attending economic hopes and fears, utopias and dystopias that are associated with the current and future development of artificial intelligence

Rooted structures in graphs: a project on Hadwiger's conjecture, rooted minors, and Tutte cycles

Hadwigers Vermutung ist eine der anspruchsvollsten Vermutungen für Graphentheoretiker und bietet eine weitreichende Verallgemeinerung des Vierfarbensatzes. Ausgehend von dieser offenen Frage der strukturellen Graphentheorie werden gewurzelte Strukturen in Graphen diskutiert. Eine Transversale einer Partition ist definiert als eine Menge, welche genau ein Element aus jeder Menge der Partition enthält und sonst nichts. Für einen Graphen G und eine Teilmenge T seiner Knotenmenge ist ein gewurzelter Minor von G ein Minor, der T als Transversale seiner Taschen enthält. Sei T eine Transversale einer Färbung eines Graphen, sodass es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T gibt; dann stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen, in T gewurzelten Minors zu gewährleisten. Diese Frage ist eng mit Hadwigers Vermutung verwoben: Eine positive Antwort würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. In dieser Arbeit wird ebendiese Fragestellung untersucht sowie weitere Konzepte vorgestellt, welche bekannte Ideen der strukturellen Graphentheorie um eine Verwurzelung erweitern. Beispielsweise wird diskutiert, inwiefern hoch zusammenhängende Teilmengen der Knotenmenge einen hoch zusammenhängenden, gewurzelten Minor erzwingen. Zudem werden verschiedene Ideen von Hamiltonizität in planaren und nicht-planaren Graphen behandelt.Hadwiger's Conjecture is one of the most tantalising conjectures for graph theorists and offers a far-reaching generalisation of the Four-Colour-Theorem. Based on this major issue in structural graph theory, this thesis explores rooted structures in graphs. A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. Given a graph G and a subset T of its vertex set, a rooted minor of G is a minor such that T is a transversal of its branch set. Assume that a graph has a transversal T of one of its colourings such that there is a system of edge-disjoint paths between all vertices from T; it comes natural to ask whether such graphs contain a minor rooted at T. This question of containment is strongly related to Hadwiger's Conjecture; indeed, a positive answer would prove Hadwiger's Conjecture for uniquely colourable graphs. This thesis studies the aforementioned question and besides, presents several other concepts of attaching rooted relatedness to ideas in structural graph theory. For instance, whether a highly connected subset of the vertex set forces a highly connected rooted minor. Moreover, several ideas of Hamiltonicity in planar and non-planar graphs are discussed