On cycles and independence in graphs

Das erste Fachkapitel ist der Berechnung von Kreispackungszahlen, d.h. der maximalen Größe kanten- bzw. eckendisjunkter Kreispackungen, gewidmet. Da diese Probleme bekanntermaßen sogar für subkubische Graphen schwer sind, behandelt der erste Abschnitt die Komplexität des Packens von Kreisen einer festen Länge l in Graphen mit Maximalgrad Delta. Dieses für l=3 von Caprara und Rizzi gelöste Problem wird hier auf alle größeren Kreislängen l verallgemeinert. Der zweite Abschnitt beschreibt die Struktur von Graphen, für die die Kreispackungszahlen einen vorgegebenen Abstand zur zyklomatischen Zahl haben. Die 2-zusammenhängenden 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 Größenordnung der minimalen Anzahl von Kreislängen 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 Kreislängen 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 Längen enthält. Der erste Abschnitt enthält eine Korrektur des ursprünglich fehlerhaften Beweises und zusätzliche Schranken. Der zweite Abschnitt leitet daraus die Unterschranke für die Anzahl der Kreislängen ab. Das letzte Fachkapitel behandelt Unterschranken für den Unabhängigkeitsquotienten, d.h. den Quotienten aus Unabhängigkeitszahl und Ordnung eines Graphen, für Graphen gegebener Dichte. In der Einleitung werden bestmögliche Schranken für die Klasse aller Graphen sowie für große zusammenhängende Graphen aus bekannten Ergebnissen abgeleitet. Danach wird die Untersuchung auf durch das Verbot kleiner ungerader Kreise eingeschränkte Graphenklassen ausgeweitet. Das Hauptergebnis des ersten Abschnitts ist eine Verallgemeinerung eines Ergebnisses von Heckman und Thomas, das die bestmögliche Schranke für zusammenhängende dreiecksfreie Graphen mit Durchschnittsgrad bis zu 10/3 impliziert und die extremalen Graphen charakterisiert. Der Rest der ersten beiden Abschnitte enthält Vermutungen ähnlichen Typs für zusammenhängende dreiecksfreie Graphen mit Durchschnittsgrad im Intervall [10/3, 54/13] und für zusammenhängende Graphen mit ungerader Taillenweite 7 mit Durchschnittsgrad bis zu 14/5. Der letzte Abschnitt enthält analoge Beobachtungen zum Bipartitionsquotienten. Die Arbeit schließt mit Vermutungen zu Unterschranken und die zugehörigen Klassen extremaler Graphen für 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

On packing shortest cycles in graphs

We study the problems to nd a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g = 3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g 2 f4; 5g, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g \geq 6, both problems are APX-hard already for graphs with maximum degree 3

Graphs with many vertex-disjoint cycles

We study graphs G in which the maximum number of vertex-disjoint cycles \nu(G) is close to the cyclomatic number \mu(G) which is a natural upper bound for \nu(G). Our main result is the existence of a finite set P(k) of graphs for all k \in \mathbb{N}_0 such that every 2-connected graph G with \mu(G) \nu (G) = k arises by applying a simple extension rule to a graph in \mathcal{P}(k). As an algorithmic consequence we describe algorithms calculating min{\mu(G)-\nu(G); k + 1} in linear time for fixed k

Chiraptophobic cockroaches evading a torch light

We propose and study game-theoretic versions of independence in graphs. The games are played by two players - the aggressor and the defender - taking alternate moves on a graph G with tokens located on vertices from an independent set of G. A move of the aggressor is to select a vertex v of G. A move of the defender is to move tokens located on vertices in NG(v) each along one incident edge. The goal of the defender is to maintain the set of occupied vertices independent while the goal of the aggressor is to make this impossible. We consider the maximum number of tokens for which the aggressor can not win in a strategic and an adaptive version of the game

On spanning tree congestion

We prove that every connected graph G of order n 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 n^{\frac{3}{2}} many edges which solves a problem posed by Ostrovskii (Minimal congestion trees, Discrete Math. 285 (2004), 219-226.

Cycle spectra of Hamiltonian graphs

AbstractWe prove that every Hamiltonian graph with n vertices and m edges has cycles with more than p−12lnp−1 different lengths, where p=m−n. For general m and n, there exist such graphs having at most 2⌈p+1⌉ different cycle lengths

Edge-injective and edge-surjective vertex labellings

For a graph G = (V,E) we consider vertex-k-labellings f : V \rightarrow {1,2,...,k} for which the induced edge weighting w : E \rightarrow {2,3,..., 2k} with w(uv) = f(u) + f(v) is injective or surjective or both. We study the relation between these labellings and the number theoretic notions of an additive basis and a Sidon set, present a new construction for a so-called restricted additive basis and derive the corresponding consequences for the labellings. We prove that a tree of order n and maximum degree \triangle has a vertex-k-labelling f for which w is bijective if and only if \triangle \leq k = n/2. Using this result we prove a recent conjecture of Ivančo and Jendrol' concerning edge-irregular total labellings for graphs that are sparse enough

Minimum degree and density of binary sequences

For $d,k\in \mathbb{N}$ with $k\leq 2d$, let $g(d,k)$ denote the infimum density of binary sequences $(x_i)_{i\in \mathbb{Z}}\in \{0,1\}^{\mathbb{Z}}$ which satisfy the minimum degree condition $\sum\limits_{j=1}^d(x_{i+j}+x_{i-j})\geq k$ for all $i\in\mathbb{Z}$ with $x_i=1$. We reduce the problem to determine $g(d,k)$ to a combinatorial problem related to the generalized $k$-girth of a graph $G$ which is defined as the minimum order of an induced subgraph of $G$ of minimum degree at least $k$. Extending results of Kézdy and Markert, and of Bermond and Peyrat, we present a minimum mean cycle formulation which allows to determine $g(d,k)$ for small values of $d$ and $k$. For odd values of $k$ with $d+1\leq k\leq 2d$, we conjecture $g(d,k)=\frac{k^2-1}{2(dk-1)}$ and show that this holds for $k\geq 2d-3$

Packing edge-disjoint cycles in graphs and the cyclomatic number

For a graph G let \mu (G) denote the cyclomatic number and let \nu (G) denote the maximum number of edge-disjoint cycles of G. We prove that for every k \geq 0 there is a nite set P(k) such that every 2-connected graph G for which \mu (G) - \nu (G) = k arises by applying a simple extension rule to a graph in P(k). Furthermore, we determine P(k) for k \leq 2 exactly

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 $3$ 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 $n$ and size $m$, our result implies the existence of an independent set of order at least $(4n-m-1)/7$