An Erdős–Gallai type theorem for uniform hypergraphs

A well-known theorem of Erdős and Gallai (1959) [1] asserts that a graph with no path of length k contains at most [Formula presented](k−1)n edges. Recently Győri et al. (2016) gave an extension of this result to hypergraphs by determining the maximum number of hyperedges in an r-uniform hypergraph containing no Berge path of length k for all values of r and k except for k=r+1. We settle the remaining case by proving that an r-uniform hypergraph with more than n edges must contain a Berge path of length r+1. © 2017 Elsevier Lt

The random k-matching-free process

Let $\mathcal{P}$ be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty $n$-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that $\mathcal{P}$ is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the $k$-matching-free process, where $\mathcal{P}$ is the property of not containing a matching of size $k$. We are able to analyse the behaviour of this process for a wide range of values of $k$; in particular we prove that if $k=o(n)$ or if $n-2k=o(\sqrt{n}/\log n)$ then this process is likely to terminate in a $k$-matching-free graph with the maximum possible number of edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds on $k$ are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime

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

On the maximum size of connected hypergraphs without a path of given length

In this note we asymptotically determine the maximum number of hyperedges possible in an $r$-uniform, connected $n$-vertex hypergraph without a Berge path of length $k$, as $n$ and $k$ tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity

Longest paths in random hypergraphs

The excellent target article of Hamm et al. (2022) raises much food for thought. In this commentary we first discuss what is included in their proposed category of ‘positive evaluations and responses to police assertions of power to attempt social influence’. Given integers k, j with 1 j k − 1, we consider the length of the longest j-tight path in the binomial random k-uniform hypergraph Hk(n, p). We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the Pathfinder algorithm, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long j-tight path

Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture

In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any $n$-vertex graph can be decomposed into $O(n)$ cycles and edges. We improve upon the previous best bound of $O(n\log\log n)$ cycles and edges due to Conlon, Fox and Sudakov, by showing an $n$-vertex graph can always be decomposed into $O(n\log^{*}n)$ cycles and edges, where $\log^{*}n$ is the iterated logarithm function.Comment: Final version, accepted for publicatio

On the structure of graphs with forbidden induced substructures

One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every $2$-edge-colouring of the complete graph on $n$ vertices there is a monochromatic clique on at least $\frac{1}{2}\log n$ vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair $(n,e)$ is the family consisting of all graphs of order $n$ and size $e$, i.e. on $n$ vertices with $e$ edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs $(m,f)$, i.e. for $n$ approaching infinity, the limit superior of the fraction of all possible sizes $e$, such that the order-size pair $(n,e)$ does not avoid the pair $(m,f)$

Colouring versus density in integers and Hales-Jewett cubes

We construct for every integer $k\geq 3$ and every real $\mu\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, \mu)$ which, when coloured with finitely many colours, contains a monochromatic $k$-term arithmetic progression, whilst every finite $Y\subseteq X$ has a subset $Z\subseteq Y$ of size $|Z|\geq \mu |Y|$ that is free of arithmetic progressions of length $k$. This answers a question of Erd\H{o}s, Ne\v{s}et\v{r}il, and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett version of this result.Comment: 5 figure