10 research outputs found
The critical window for the classical Ramsey-Tur\'an problem
The first application of Szemer\'edi's powerful regularity method was the
following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any
K_4-free graph on N vertices with independence number o(N) has at most (1/8 +
o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising
geometric construction, utilizing the isoperimetric inequality for the high
dimensional sphere, of a K_4-free graph on N vertices with independence number
o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in
1976, several problems have been asked on estimating the minimum possible
independence number in the critical window, when the number of edges is about
N^2 / 8. These problems have received considerable attention and remained one
of the main open problems in this area. In this paper, we give nearly
best-possible bounds, solving the various open problems concerning this
critical window.Comment: 34 page
Independence, odd girth, and average degree
We prove several best-possible lower bounds in terms of the order and the average degree for the independence number of graphs which are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most due to Heckman and Thomas [A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, {\it Discrete Math.} {\bf 233} (2001), 233-237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order and size , our result implies the existence of an independent set of order at least
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
In the complement of a dominating set
A set D of vertices of a graph G=(V,E) is a dominating set, if every vertex
of D\V has at least one neighbor that belongs to D. The disjoint domination
number of a graph G is the minimum cardinality of two disjoint dominating
sets of G. We prove upper bounds for the disjoint domination number for
graphs of minimum degree at least 2, for graphs of large minimum degree and
for cubic graphs.A set T of vertices of a graph G=(V,E) is a total
dominating set, if every vertex of G has at least one neighbor that belongs
to T. We characterize graphs of minimum degree 2 without induced 5-cycles
and graphs of minimum degree at least 3 that have a dominating set, a total
dominating set, and a non-empty vertex set that are disjoint.A set I of
vertices of a graph G=(V,E) is an independent set, if all vertices in I are
not adjacent in G. We give a constructive characterization of trees that
have a maximum independent set and a minimum dominating set that are
disjoint and we show that the corresponding decision problem is NP-hard for
general graphs. Additionally, we prove several structural and hardness
results concerning pairs of disjoint sets in graphs which are dominating,
independent, or both. Furthermore, we prove lower bounds for the maximum
cardinality of an independent set of graphs with specifed odd girth and
small average degree.A connected graph G has spanning tree congestion at
most s, if G has a spanning tree T such that for every edge e of T the edge
cut defined in G by the vertex sets of the two components of T-e contains
at most s edges. We prove that every connected graph of order n has
spanning tree congestion at most n^(3/2) and we show that the corresponding
decision problem is NP-hard