How tough is toughness?

The concept of toughness was introduced by Chvátal [34] more than forty years ago. Toughness resembles vertex connectivity, but is different in the sense that it takes into account what the effect of deleting a vertex cut is on the number of resulting components. As we will see, this difference has major consequences in terms of computational complexity and on the implications with respect to cycle structure, in particular the existence of Hamilton cycles and k-factors

A look at cycles containing specified elements of a graph

AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration

Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to the first problems shown to be computationally hard when the theory of NP-completeness was being developed. A lot of research has been devoted to the complexity of Hamiltonian path and Hamiltonian cycle problems for special graph classes, yet only a handful of positive results are known. The complexities of both of these problems have been open even for $4K_1$-free graphs, i.e., graphs of independence number at most $3$. We answer this question in the general setting of graphs of bounded independence number. We also consider a newly introduced problem called \emph{Hamiltonian-$\ell$-Linkage} which is related to the notions of a path cover and of a linkage in a graph. This problem asks if given $\ell$ pairs of vertices in an input graph can be connected by disjoint paths that altogether traverse all vertices of the graph. For $\ell=1$, Hamiltonian-1-Linkage asks for existence of a Hamiltonian path connecting a given pair of vertices. Our main result reads that for every pair of integers $k$ and $\ell$, the Hamiltonian-$\ell$-Linkage problem is polynomial time solvable for graphs of independence number not exceeding $k$. We further complement this general polynomial time algorithm by a structural description of obstacles to Hamiltonicity in graphs of independence number at most $k$ for small values of $k$

A Characterization of Uniquely Representable Graphs

The betweenness structure of a finite metric space M =(X, d) is a pair ℬ (M)=(X, βM) where βM is the so-called betweenness relation of M that consists of point triplets (x, y, z) such that d(x, z)= d(x, y)+ d(y, z). The underlying graph of a betweenness structure ℬ =(X, β)isthe simple graph G(ℬ)=(X, E) where the edges are pairs of distinct points with no third point between them. A connected graph G is uniquely representable if there exists a unique metric betweenness structure with underlying graph G. It was implied by previous works that trees are uniquely representable. In this paper, we give a characterization of uniquely representable graphs by showing that they are exactly the block graphs. Further, we prove that two related classes of graphs coincide with the class of block graphs and the class of distance-hereditary graphs, respectively. We show that our results hold not only for metric but also for almost-metric betweenness structures. © 2021 Péter G.N. Szabó

Bounds for the independence number of a graph

The independence number of a graph is the maximum number of vertices from the vertex set of the graph such that no two vertices are adjacent. We systematically examine a collection of upper bounds for the independence number to determine graphs for which each upper bound is better than any other upper bound considered. A similar investigation follows for lower bounds. In several instances a graph cannot be found. We also include graphs for which no bound equals $\alpha$ and bounds which do not apply to general graphs

Extremal Graph Theory: Basic Results

Η παρούσα διπλωματική εργασία έχει σκοπό να παρουσιάσει μία σφαιρική εικόνα της θεωρίας των ακραίων γραφημάτων, διερευνώντας κοινές τεχνικές και τον τρόπο που εφαρμόζονται σε κάποια από τα πιο διάσημα αποτελέσματα του τομέα. Το πρώτο κεφάλαιο είναι μία εισαγωγή στο θέμα και κάποιοι προαπαιτούμενοι ορισμοί και αποτελέσματα. Το δεύτερο κεφάλαιο αφορά υποδομές πυκνών γραφημάτων και εστιάζει σε σημαντικά αποτελέσματα όπως είναι το θεώρημα του Turán, το λήμμα κανονικότητας του Szemerédi και το θεώρημα των Erdős-Stone-Simonovits. Το τρίτο κεφάλαιο αφορά υποδομές αραιών γραφημάτων και ερευνά συνθήκες που εξαναγκάζουν ένα γράφημα που περιέχει ένα δοθέν έλασσον ή τοπολογικό έλασσον. Το τέταρτο και τελευταίο κεφάλαιο είναι μία εισαγωγή στην θεωρία ακραίων r-ομοιόμορφων υπεργραφημάτων και περιέχει αποτελέσματα που αφορούν συνθήκες οι οποίες τα εξαναγκάζουν να περιέχουν πλήρη r-γραφήματα και Χαμιλτονιανούς κύκλους.In this thesis, we take a general overview of extremal graph theory, investigating common techniques and how they apply to some of the more celebrated results in the field. The first chapter is an introduction to the subject and some preliminary definitions and results. The second chapter concerns substructures in dense graphs and focuses on important results such as Turán’s theorem, Szemerédi’s regularity lemma and the Erdős-Stone-Simonovits theorem. The third chapter concerns substructures in sparse graphs and investigates conditions which force a graph to contain a certain minor or topological minor. The fourth and final chapter is an introduction to the extremal theory of r-uniform hypergraphs and consists of a presentation of results concerning the conditions which force them to contain a complete r-graph and a Hamiltonian cycle