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Small vertex-transitive graphs of given degree and girth
We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k, g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pairof parameters k â„ 2 and g â„ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs forcertain families with prime power girth, and generalize the construction of Bray, Parker and Rowley that has yielded several of the smallest known (k, g)-graphs
On cycles and independence in graphs
ï»żDas erste Fachkapitel ist der Berechnung von Kreispackungszahlen, d.h. der maximalen GröĂe kanten- bzw. eckendisjunkter Kreispackungen, gewidmet. Da diese Probleme bekanntermaĂen sogar fĂŒr subkubische Graphen schwer sind, behandelt der erste Abschnitt die KomplexitĂ€t des Packens von Kreisen einer festen LĂ€nge l in Graphen mit Maximalgrad Delta. Dieses fĂŒr l=3 von Caprara und Rizzi gelöste Problem wird hier auf alle gröĂeren KreislĂ€ngen l verallgemeinert. Der zweite Abschnitt beschreibt die Struktur von Graphen, fĂŒr die die Kreispackungszahlen einen vorgegebenen Abstand zur zyklomatischen Zahl haben. Die 2-zusammenhĂ€ngenden Graphen mit dieser Eigenschaft können jeweils durch Anwendung einer einfachen Erweiterungsregel auf eine endliche Menge von Graphen erzeugt werden. Aus diesem Strukturergebnis wird ein fpt-Algorithmus abgeleitet.
Das zweite Fachkapitel handelt von der GröĂenordnung der minimalen Anzahl von KreislĂ€ngen in einem Hamiltongraph mit q Sehnen. Eine Familie von Beispielen zeigt, dass diese Unterschranke höchstens die Wurzel von q+1 ist. Dem Hauptsatz dieses Kapitels zufolge ist die Zahl der KreislĂ€ngen eines beliebigen Hamiltongraphen mit q Sehnen mindestens die Wurzel von 4/7*q. Der Beweis beruht auf einem Lemma von Faudree et al., demzufolge der Graph, der aus einem Weg mit Endecken x und y und q gleichlangen Sehnen besteht, x-y-Wege von mindestens q/3 verschiedenen LĂ€ngen enthĂ€lt. Der erste Abschnitt enthĂ€lt eine Korrektur des ursprĂŒnglich fehlerhaften Beweises und zusĂ€tzliche Schranken. Der zweite Abschnitt leitet daraus die Unterschranke fĂŒr die Anzahl der KreislĂ€ngen ab.
Das letzte Fachkapitel behandelt Unterschranken fĂŒr den UnabhĂ€ngigkeitsquotienten, d.h. den Quotienten aus UnabhĂ€ngigkeitszahl und Ordnung eines Graphen, fĂŒr Graphen gegebener Dichte. In der Einleitung werden bestmögliche Schranken fĂŒr die Klasse aller Graphen sowie fĂŒr groĂe zusammenhĂ€ngende Graphen aus bekannten Ergebnissen abgeleitet. Danach wird die Untersuchung auf durch das Verbot kleiner ungerader Kreise eingeschrĂ€nkte Graphenklassen ausgeweitet. Das Hauptergebnis des ersten Abschnitts ist eine Verallgemeinerung eines Ergebnisses von Heckman und Thomas, das die bestmögliche Schranke fĂŒr zusammenhĂ€ngende dreiecksfreie Graphen mit Durchschnittsgrad bis zu 10/3 impliziert und die extremalen Graphen charakterisiert. Der Rest der ersten beiden Abschnitte enthĂ€lt Vermutungen Ă€hnlichen Typs fĂŒr zusammenhĂ€ngende dreiecksfreie Graphen mit Durchschnittsgrad im Intervall [10/3, 54/13] und fĂŒr zusammenhĂ€ngende Graphen mit ungerader Taillenweite 7 mit Durchschnittsgrad bis zu 14/5. Der letzte Abschnitt enthĂ€lt analoge Beobachtungen zum Bipartitionsquotienten. Die Arbeit schlieĂt mit Vermutungen zu Unterschranken und die zugehörigen Klassen extremaler Graphen fĂŒr den Bipartitionsquotienten.This thesis discusses several problems related to cycles and the independence number in graphs.
Chapter 2 contains problems on independent sets of cycles. It is known that it is hard to compute the maximum cardinality of edge-disjoint and vertex-disjoint cycle packings, even if restricted to subcubic graphs. Therefore, the first section discusses the complexity of a simpler problem: packing cycles of fixed length l in graphs of maximum degree Delta. The results of Caprara and Rizzi, who have solved this problem for l=3 are generalised to arbitrary lengths. The second section describes the structure of graphs for which the edge-disjoint resp. vertex-disjoint cycle packing number differs from the cyclomatic number by a constant. The corresponding classes of 2-connected graphs can be obtained by a simple extension rules applied to a finite set of graphs. This result implies a fixed-parameter-tractability result for the edge-disjoint and vertex-disjoint cycle packing numbers.
Chapter 3 contains an approximation of the minimum number of cycle lengths in a Hamiltonian graph with q chords. A family of examples shows that no more than the square root of q+1 can be guaranteed. The main result is that the square root of 4/7*q cycle lengths can be guaranteed. The proof relies on a lemma by Faudree et al.,
which states that the graph that contains a path with endvertices x and y and q chords of equal length contains paths between x and y of at least q/3 different lengths. The first section corrects the originally faulty proof and derives additional bounds. The second section uses these bounds to derive the lower bound on the size of the cycle spectrum.
Chapter 4 focuses on lower bounds on the independence ratio, i.e. the quotient of independence number and order of a graph, for graphs of given density. In the introduction, best-possible bounds both for arbitrary graphs and large connected graphs are derived from known results. Therefore, the rest of this chapter considers classes of graphs defined by forbidding small odd cycles as subgraphs. The main result of the first section is a generalisation of a result of Heckman and Thomas that determines the best possible lower bound for connected triangle-free graphs with average degree at most 10/3
and characterises the extremal graphs. The rest of the chapter is devoted to conjectures with similar statements on connected triangle-free graphs of average degree in [10/3, 54/13] and on connected graphs of odd girth 7 with average degree up to 14/5, similar conjectures for the bipartite ratio, possible classes of extremal graphs for these conjectures, and observations in support of the conjectures
Two problems in graph theory
In the thesis we study two topics in graph theory. The first one is concerned with the famous conjecture of Hadwiger that every graph G without a minor of a complete graph on t +1 vertices can be coloured with t colours. We investigate how large an induced subgraph of G can be, so that the subgraph can be coloured with t colours. We show that G admits a t-colourable induced subgraph on more than half of its vertices. Moreover, if such graph G on n vertices does not contain any triangle, we show it admits a t-colourable induced subgraph on at least 4n=5 vertices and show even better bounds for graphs with larger odd girth. The second topic is a variant of a well-known two player Maker-Breaker connectivity game in which players take turns choosing an edge in each step in order to achieve their respective goals. While a complete characterisation is known for the connectivity game in which both players choose a single edge, much less is known in all other cases. We study the variant in which both players choose two edges, or more generally, the variant in which the first player decides whether both players choose one or two edges in the next round
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction
Hyperbolicity of direct products of graphs
It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G(1) x G(2) is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).This work was supported in part by four grants from Ministerio de EconomĂa y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain