Colorings of graphs, digraphs, and hypergraphs

Brooks' Theorem ist eines der bekanntesten Resultate über Graphenfärbungen: Sei G ein zusammenhängender Graph mit Maximalgrad d. Ist G kein vollständiger Graph, so lassen sich die Ecken von G so mit d Farben färben, dass zwei benachbarte Ecken unterschiedlich gefärbt sind. In der vorliegenden Arbeit liegt der Fokus auf Verallgemeinerungen von Brooks Theorem für Färbungen von Hypergraphen und gerichteten Graphen. Eine Färbung eines Hypergraphen ist eine Färbung der Ecken so, dass keine Kante monochromatisch ist. Auf Hypergraphen erweitert wurde der Satz von Brooks von R.P. Jones. Im ersten Teil der Dissertation werden Möglichkeiten aufgezeigt, das Resultat von Jones weiter zu verallgemeinern. Kernstück ist ein Zerlegungsresultat: Zu einem Hypergraphen H und einer Folge f=(f_1,…,f_p) von Funktionen, welche von V(H) in die natürlichen Zahlen abbilden, wird untersucht, ob es eine Zerlegung von H in induzierte Unterhypergraphen H_1,…,H_p derart gibt, dass jedes H_i strikt f_i-degeneriert ist. Dies bedeutet, dass jeder Unterhypergraph H_i' von H_i eine Ecke v enthält, deren Grad in H_i' kleiner als f_i(v) ist. Es wird bewiesen, dass die Bedingung f_1(v)+…+f_p(v) \geq d_H(v) für alle v fast immer ausreichend für die Existenz einer solchen Zerlegung ist und gezeigt, dass sich die Ausnahmefälle gut charakterisieren lassen. Durch geeignete Wahl der Funktion f lassen sich viele bekannte Resultate ableiten, was im dritten Kapitel erörtert wird. Danach werden zwei weitere Verallgemeinerungen des Satzes von Jones bewiesen: Ein Theorem zu DP-Färbungen von Hypergraphen und ein Resultat, welches die chromatische Zahl eines Hypergraphen mit dessen maximalem lokalen Kantenzusammenhang verbindet. Der zweite Teil untersucht Färbungen gerichteter Graphen. Eine azyklische Färbung eines gerichteten Graphen ist eine Färbung der Eckenmenge des gerichteten Graphen sodass es keine monochromatischen gerichteten Kreise gibt. Auf dieses Konzept lassen sich viele klassische Färbungsresultate übertragen. Dazu zählt auch Brooks Theorem, wie von Mohar bewiesen wurde. Im siebten Kapitel werden DP-Färbungen gerichteter Graphen untersucht. Insbesondere erfolgt der Transfer von Mohars Theorem auf DP-Färbungen. Das darauffolgende Kapitel befasst sich mit kritischen gerichteten Graphen. Insbesondere werden Konstruktionen für diese angegeben und die gerichtete Version des Satzes von Hajós bewiesen.Brooks‘ Theorem is one of the most known results in graph coloring theory: Let G be a connected graph with maximum degree d >2. If G is not a complete graph, then there is a coloring of the vertices of G with d colors such that no two adjacent vertices get the same color. Based on Brooks' result, various research topics in graph coloring arose. Also, it became evident that Brooks' Theorem could be transferred to many other coloring-concepts. The present thesis puts its focus especially on two of those concepts: hypergraphs and digraphs. A coloring of a hypergraph H is a coloring of its vertices such that no edge is monochromatic. Brooks' Theorem for hypergraphs was obtained by R.P. Jones. In the first part of this thesis, we present several ways how to further extend Jones' theorem. The key element is a partition result, to which the second chapter is devoted. Given a hypergraph H and a sequence f=(f_1,…,f_p) of functions, we examine if there is a partition of $H$ into induced subhypergraphs H_1,…,H_p such that each of the H_i is strictly f_i-degenerate. This means that in each non-empty subhypergraph H_i' of H_i there is a vertex v having degree d_{H_i'}(v

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol. IV

The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors - all collected in the previous three decades (1980-2010) – but most of them are from the last decade (2000-2010), some of them being lost and found, yet others are extended, diversified, improved versions. This is an eclectic tome of 800 pages with papers in various fields of sciences, alphabetically listed, such as: astronomy, biology, calculus, chemistry, computer programming codification, economics and business and politics, education and administration, game theory, geometry, graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, psychology, quantum physics, scientific research methods, and statistics. It was my preoccupation and collaboration as author, co-author, translator, or cotranslator, and editor with many scientists from around the world for long time. Many topics from this book are incipient and need to be expanded in future explorations

Annual report 2015

Accessions considered in the study. Overview of the material considered in this study. For all materials, the GenBank identifier, the accession and species name as used in this study (Species) as well as their species synonyms used in the donor seed banks or in the NCBI GenBank (Material source/Reference) are provided. The genome symbol, and the country of origin, where the material was originally collected are given. The ploidy level measured in the scope of this study and the information if a herbarium voucher could be deposited in the herbarium of IPK Gatersleben (GAT) is given. Genomic formulas of tetraploids and hexploids are given as “female x male parent”. The genomes of Aegilops taxa follow Kilian et al. [74] and Li et al. [84]. Genome denominations for Hordeum follow Blattner [107] and Bernhardt [12] for the remaining taxa. (XLS 84 kb

The complexity of the Hajós calculus for planar graphs

The planar Haj´os calculus is the Haj´os calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. We prove that the planar Haj´os calculus is polynomially bounded iff the Haj´os calculus is polynomially bounded

Rented worlds: bedsits, boarding houses and multiple occupancy homes in postwar London, 1945-1963

This thesis explores the history of a series of overlooked yet thoroughly commonplace domestic spaces, through an interdisciplinary approach combining visual culture and archival research. Having once housed the overwhelming majority of people in Britain, private rented accommodation entered a period of decline that accelerated after the Second World War. Yet so large was the nineteenth century inheritance embodied by the private landlord that such dwellings still housed more than any other tenure category well into the 1950s. The way people experienced these changes varied greatly. Factors of race, class and gender were refracted through the geography of the city, as the concentration within London of surviving ‘residues’ of private renting magnified their social significance. Often referred to simply as ‘rented rooms’, such housing encompassed a variety of different types, from working-class lodging houses and multiple occupancy homes, to middle-class boarding houses and residential hotels. Together they provided the setting for a whole host of ‘social problems’, including issues of public safety and the need to overcome the Victorian legacy of ‘squalor’; the disassembling effects upon families and communities of ‘social disorganisation’; the dubitable standing of the rentier class during a period of national reconstruction; and fears over ghettoization and the challenges that Commonwealth immigration posed to Britain’s cultural identity. A declining private rental sector formed the connecting matrix for these distinctive postwar problems. The regulatory endeavours of the welfare state sought to penetrate these spaces precisely because of their seeming obscurity. At the same time, the ruinous state of London’s rented worlds sheltered forms of life that would not have been possible elsewhere, while also promising opportunities to property speculators. Three chosen themes, centred on fire safety, loneliness and landlordism, shed light on how these issues were contested from multiple angles by municipal government, social investigators, developers, and tenants themselves. The thesis argues that the centrality of the housing question to the development of the postwar welfare state – insofar as this went beyond the reconstruction of the built environment to encompass the regulation of domestic life – took its bearings in significant part from the overlooked spaces of London’s ‘rented rooms’. State regulation ran up against questions of ownership and urban capital, as the Conservative vision of a ‘property owning democracy’ sought to unpick the legacy of rent control, while the same period witnessed the rise of gentrification amidst Labour policies that curtailed the rights of property. Grasping how the changing status of property meshes with the politics of domestic and urban space opens up a rich field of materials – including popular films, architectural exhibits, cartoons, maps, valuation lists and testimonies from rent tribunals – that can deepen our understanding of the postwar period