14 research outputs found

    The Lov\'asz-Cherkassky theorem in infinite graphs

    Full text link
    Infinite generalizations of theorems in finite combinatorics were initiated by Erd\H{o}s due to his famous Erd\H{o}s-Menger conjecture (now known as the Aharoni-Berger theorem) that extends Menger's theorem to infinite graphs in a structural way. We prove a generalization of this manner of the classical result about packing edge-disjoint T T -paths in an ``inner Eulerian'' setting obtained by Lov\'asz and Cherkassky independently in the '70s

    Pancyclicity of Hamiltonian graphs

    Full text link
    An nn-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from 33 up to nn. In 1972, Erd\H{o}s conjectured that every Hamiltonian graph with independence number at most kk and at least n=Ω(k2)n = \Omega(k^2) vertices is pancyclic. In this paper we prove this old conjecture in a strong form by showing that if such a graph has n=(2+o(1))k2n = (2+o(1))k^2 vertices, it is already pancyclic, and this bound is asymptotically best possible.Comment: 12 pages, 3 figure

    Colorings of graphs, digraphs, and hypergraphs

    Get PDF
    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 HH 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

    Packings and coverings with Hamilton cycles and on-line Ramsey theory

    Get PDF
    A major theme in modern graph theory is the exploration of maximal packings and minimal covers of graphs with subgraphs in some given family. We focus on packings and coverings with Hamilton cycles, and prove the following results in the area. • Let ε > 0, and let GG be a large graph on n vertices with minimum degree at least (1=2+ ε)n. We give a tight lower bound on the size of a maximal packing of GG with edge-disjoint Hamilton cycles. • Let TT be a strongly k-connected tournament. We give an almost tight lower bound on the size of a maximal packing of TT with edge-disjoint Hamilton cycles. • Let log 1^11^17^7 nn/nn≤pp≤1-nn−^-1^1/^/8^8. We prove that GGn_n,_,p_p may a.a.s be covered by a set of ⌈Δ(GGn_n,_,p_p)/2⌉ Hamilton cycles, which is clearly best possible. In addition, we consider some problems in on-line Ramsey theory. Let r(GG,HH) denote the on-line Ramsey number of GG and HH. We conjecture the exact values of r (PPk_k,PPℓ_ℓ) for all kk≤ℓ. We prove this conjecture for kk=2, prove it to within an additive error of 10 for kk=3, and prove an asymptotically tight lower bound for kk=4. We also determine r(PP3_3,CCℓ_ℓ exactly for all ℓ

    Subject Index Volumes 1–200

    Get PDF

    Network based data oriented methods for application driven problems

    Get PDF
    Networks are amazing. If you think about it, some of them can be found in almost every single aspect of our life from sociological, financial and biological processes to the human body. Even considering entities that are not necessarily connected to each other in a natural sense, can be connected based on real life properties, creating a whole new aspect to express knowledge. A network as a structure implies not only interesting and complex mathematical questions, but the possibility to extract hidden and additional information from real life data. The data that is one of the most valuable resources of this century. The different activities of the society and the underlying processes produces a huge amount of data, which can be available for us due to the technological knowledge and tools we have nowadays. Nevertheless, the data without the contained knowledge does not represent value, thus the main focus in the last decade is to generate or extract information and knowledge from the data. Consequently, data analytics and science, as well as data-driven methodologies have become leading research fields both in scientific and industrial areas. In this dissertation, the author introduces efficient algorithms to solve application oriented optimization and data analysis tasks built on network science based models. The main idea is to connect these problems along graph based approaches, from virus modelling on an existing system through understanding the spreading mechanism of an infection/influence and maximize or minimize the effect, to financial applications, such as fraud detection or cost optimization in a case of employee rostering

    Algorithms and Models for the Web Graph

    Get PDF
    corecore