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

Digraph Colouring and Arc-Connectivity

The dichromatic number $\vec\chi(D)$ of a digraph $D$ is the minimum size of a partition of its vertices into acyclic induced subgraphs. We denote by $\lambda(D)$ the maximum local edge connectivity of a digraph $D$. Neumann-Lara proved that for every digraph $D$, $\vec\chi(D) \leq \lambda(D) + 1$. In this paper, we characterize the digraphs $D$ for which $\vec\chi(D) = \lambda(D) + 1$. This generalizes an analogue result for undirected graph proved by Stiebitz and Toft as well as the directed version of Brooks' Theorem proved by Mohar. Along the way, we introduce a generalization of Haj\'os join that gives a new way to construct families of dicritical digraphs that is of independent interest.Comment: 34 pages, 11 figure

The Resolution of Keller's Conjecture

We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Keller's conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $\mathbb{R}^7$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $\mathbb{R}^8$ exists (which we also verify), this completely resolves Keller's conjecture.Comment: 25 pages, 9 figures, 3 tables; IJCAR 202

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

Novel procedures for graph edge-colouring

Orientador: Dr. Renato CarmoCoorientador: Dr. André Luiz Pires GuedesTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa : Curitiba, 05/12/2018Inclui referências e índiceÁrea de concentração: Ciência da ComputaçãoResumo: O índice cromático de um grafo G é o menor número de cores necessário para colorir as arestas de G de modo que não haja duas arestas adjacentes recebendo a mesma cor. Pelo célebre Teorema de Vizing, o índice cromático de qualquer grafo simples G ou é seu grau máximo , ou é ? + 1, em cujo caso G é dito Classe 1 ou Classe 2, respectivamente. Computar uma coloração de arestas ótima de um grafo ou simplesmente determinar seu índice cromático são problemas NP-difíceis importantes que aparecem em aplicações notáveis, como redes de sensores, redes ópticas, controle de produção, e jogos. Neste trabalho, nós apresentamos novos procedimentos de tempo polinomial para colorir otimamente as arestas de grafos pertences a alguns conjuntos grandes. Por exemplo, seja X a classe dos grafos cujos maiorais (vértices de grau ?) possuem soma local de graus no máximo ?2 ?? (entendemos por 'soma local de graus' de um vértice x a soma dos graus dos vizinhos de x). Nós mostramos que quase todo grafo está em X e, estendendo o procedimento de recoloração que Vizing usou na prova para seu teorema, mostramos que todo grafo em X é Classe 1. Nós também conseguimos resultados em outras classes de grafos, como os grafos-junção, os grafos arco-circulares, e os prismas complementares. Como um exemplo, nós mostramos que um prisma complementar só pode ser Classe 2 se for um grafo regular distinto do K2. No que diz respeito aos grafos-junção, nós mostramos que se G1 e G2 são grafos disjuntos tais que |V(G1)| _ |V(G2)| e ?(G1) _ ?(G2), e se os maiorais de G1 induzem um grafo acíclico, então o grafo-junção G1 ?G2 é Classe 1. Além desses resultados em coloração de arestas, apresentamos resultados parciais em coloração total de grafos-junção, de grafos arco-circulares, e de grafos cobipartidos, bem como discutimos um procedimento de recoloração para coloração total. Palavras-chave: Coloração de grafos e hipergrafos (MSC 05C15). Algoritmos de grafos (MSC 05C85). Teoria dos grafos em relação à Ciência da Computação (MSC 68R10). Graus de vértices (MSC 05C07). Operações de grafos (MSC 05C76).Abstract: The chromatic index of a graph G is the minimum number of colours needed to colour the edges of G in a manner that no two adjacent edges receive the same colour. By the celebrated Vizing's Theorem, the chromatic index of any simple graph G is either its maximum degree ? or it is ? + 1, in which case G is said to be Class 1 or Class 2, respectively. Computing an optimal edge-colouring of a graph or simply determining its chromatic index are important NP-hard problems which appear in noteworthy applications, like sensor networks, optical networks, production control, and games. In this work we present novel polynomial-time procedures for optimally edge-colouring graphs belonging to some large sets of graphs. For example, let X be the class of the graphs whose majors (vertices of degree ?) have local degree sum at most ?2 ? ? (by 'local degree sum' of a vertex x we mean the sum of the degrees of the neighbours of x). We show that almost every graph is in X and, by extending the recolouring procedure used by Vizing's in the proof for his theorem, we show that every graph in X is Class 1. We further achieve results in other graph classes, such as join graphs, circular-arc graphs, and complementary prisms. For instance, we show that a complementary prism can be Class 2 only if it is a regular graph distinct from the K2. Concerning join graphs, we show that if G1 and G2 are disjoint graphs such that |V(G1)| _ |V(G2)| and ?(G1) _ ?(G2), and if the majors of G1 induce an acyclic graph, then the join graph G1 ?G2 is Class 1. Besides these results on edge-colouring, we present partial results on total colouring join graphs, cobipartite graphs, and circular-arc graphs, as well as a discussion on a recolouring procedure for total colouring. Keywords: Colouring of graphs and hypergraphs (MSC 05C15). Graph algorithms (MSC 05C85). Graph theory in relation to Computer Science (MSC 68R10). Vertex degrees (MSC 05C07). Graph operations (MSC 05C76)

Signed graphs, regular matroids, grafts

Mathematics

Convex Algebraic Geometry Approaches to Graph Coloring and Stable Set Problems

The objective of a combinatorial optimization problem is to find an element that maximizes a given function defined over a large and possibly high-dimensional finite set. It is often the case that the set is so large that solving the problem by inspecting all the elements is intractable. One approach to circumvent this issue is by exploiting the combinatorial structure of the set (and possibly the function) and reformulate the problem into a familiar set-up where known techniques can be used to attack the problem. Some common solution methods for combinatorial optimization problems involve formulations that make use of Systems of Linear Equations, Linear Programs (LPs), Semidefinite Programs (SDPs), and more generally, Conic and Semi-algebraic Programs. Although, generality often implies flexibility and power in the formulations, in practice, an increase in sophistication usually implies a higher running time of the algorithms used to solve the problem. Despite this, for some combinatorial problems, it is hard to rule out the applicability of one formulation over the other. One example of this is the Stable Set Problem. A celebrated result of Lovász's states that it is possible to solve (to arbitrary accuracy) in polynomial time the Stable Set Problem for perfect graphs. This is achieved by showing that the Stable Set Polytope of a perfect graph is the projection of a slice of a Positive Semidefinite Cone of not too large dimension. Thus, the Stable Set Problem can be solved with the use of a reasonably sized SDP. However, it is unknown whether one can solve the same problem using a reasonably sized LP. In fact, even for simple classes of perfect graphs, such as Bipartite Graphs, we do not know the right order of magnitude of the minimum size LP formulation of the problem. Another example is Graph Coloring. In 2008 Jesús De Loera, Jon Lee, Susan Margulies and Peter Malkin proposed a technique to solve several combinatorial problems, including Graph Coloring Problems, using Systems of Linear Equations. These systems are obtained by reformulating the decision version of the combinatorial problem with a system of polynomial equations. By a theorem of Hilbert, known as Hilbert's Nullstellensatz, the infeasibility of this polynomial system can be determined by solving a (usually large) system of linear equations. The size of this system is an exponential function of a parameter $d$ that we call the degree of the Nullstellensatz Certificate. Computational experiments of De Loera et al. showed that the Nullstellensatz method had potential applications for detecting non-$3$-colorability of graphs. Even for known hard instances of graph coloring with up to two thousand vertices and tens of thousands of edges the method was useful. Moreover, all of these graphs had very small Nullstellensatz Certificates. Although, the existence of hard non-$3$-colorable graph examples for the Nullstellensatz approach are known, determining what combinatorial properties makes the Nullstellensatz approach effective (or ineffective) is wide open. The objective of this thesis is to amplify our understanding on the power and limitations of these methods, all of these falling into the umbrella of Convex Algebraic Geometry approaches, for combinatorial problems. We do this by studying the behavior of these approaches for Graph Coloring and Stable Set Problems. First, we study the Nullstellensatz approach for graphs having large girth and chromatic number. We show that that every non-$k$-colorable graph with girth $g$ needs a Nullstellensatz Certificate of degree $\Omega(g)$ to detect its non-$k$-colorability. It is our general belief that the power of the Nullstellensatz method is tied with the interplay between local and global features of the encoding polynomial system. If a graph is locally $k$-colorable, but globally non-$k$-colorable, we suspect that it will be hard for the Nullstellensatz to detect the non-$k$-colorability of the graph. Our results point towards that direction. Finally, we study the Stable Set Problem for $d$-regular Bipartite Graphs having no $C_4$, i.e., having no cycle of length four. In 2017 Manuel Aprile \textit{et al.} showed that the Stable Set Polytope of the incidence graph $G_{d-1}$ of a Finite Projective Plane of order $d-1$ (hence, $d$-regular) does not admit an LP formulation with fewer than $\frac{\ln(d)}{d}|E(G_{d-1})|$ facets. Although, we did not manage to improve this lower bound for general $d$-regular graphs, we show that any $4$-regular bipartite graph $G$ having no $C_4$ does not admit an LP formulation with fewer than $|E(G)|$ facets. In addition, we obtain computational results showing the $|E(G)|$ lower bound also holds for the Finite Projective Plane $G_4$, a $5$-regular graph. It is our belief that Aprile et al. bounds can be improved considerably

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Polynomial growth of concept lattices, canonical bases and generators:: extremal set theory in Formal Concept Analysis

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts