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Hamiltonian Cycles in Certain Graphs and Out-Arc Pancyclic Vertices in Tournaments

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Hamiltonian Cycles in Certain Graphs and Out-Arc Pancyclic Vertices in Tournaments Hamiltonian Cycles in Certain Graphs and Out-arc Pancyclic Vertices in Tournaments Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der Rheinisch- Westf¨alischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Master of Science Jinfeng Feng aus Shanxi, V.R. China Berichter: Prof. Dr. Yubao Guo Prof. Dr. Eberhard Triesch Tag der m¨undlichen Pr¨ufung: 24. 01. 2008 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verf¨ugbar. AACHENER BEITRÄGE ZUR MATHEMATIK Herausgeber: Professor Dr. H. H. Bock, Institut für Statistik und Wirtschaftsmathematik Professor Dr. Dr. h.c. H. Th. Jongen, Lehrstuhl C für Mathematik Professor Dr. W. Plesken, Lehrstuhl B für Mathematik Feng, Jinfeng Hamiltonian Cycles in Certain Graphs and Out-arc Pancyclic Vertices in Tournaments ISBN 3-86130-136-9 Bibliografische Informationen der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.ddb.de abrufbar. Das Werk, einschließlich seiner Teile, ist urheberrechtlich geschützt. Jede Verwendung ist ohne die Zustimmung des Herausgebers außerhalb der engen Grenzen des Urhebergesetzes unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Vertrieb: 1. Auflage 2008 © Verlagshaus Mainz GmbH Aachen Süsterfeldstr. 83, 52072 Aachen Tel. 0241/87 34 34 Fax 0241/87 55 77 www.Verlag-Mainz.de Herstellung: Druck und Verlagshaus Mainz GmbH Aachen Süsterfeldstr. 83, 52072 Aachen Tel. 0241/87 34 34 Fax 0241/87 55 77 www.Verlag-Mainz.de www.DruckereiMainz.de www.Druckservice-Aachen.de Satz: nach Druckvorlage des Autors Umschlaggestaltung: Druckerei Mainz printed in Germany Gedruckt auf chlorfrei gebleichtem Papier Preface Graph theory is a delightful research area in discrete mathematics, and its results have many applications in computer science, social and natural science. The Hamiltonian problem is concerned with conditions under which a graph contains a spanning cycle. Named after Sir William Rowan Hamilton, this problem traces its origins to the 1850s. Although this problem is NP-complete, in the last 100 years many sufficient conditions for a graph to be Hamiltonian have been found and many related topics have also been discussed, for example, pancyclicity, proper colored Hamiltonian paths or cycles, etc. In the first part of this thesis, we introduce new sufficient conditions for a graph to be Hamiltonian and some other results on related topics. Chapter 1 will be devoted to an introduction of the terminology and some well-known results about Hamiltonian graphs. Generally speaking, there are two important types of sufficiency conditions with re- spect to the cyclic properties of graphs. The first is based on the so-called numerical conditions, of which probably degree conditions are the most well-known; structural con- ditions form the second important type. A typical example of the latter is forbidden subgraph conditions. In Chapter 2, we give some new sufficient conditions for a graph to be Hamiltonian by combining those two types. Let G = (V, E) be a connected graph. The distance between two vertices x and y in G, denoted by dG(x, y), is the length of a shortest path between x and y. A graph G is called almost distance-hereditary if each connected induced subgraph H of G has the property that dH (u, v) ≤ dG(u, v) + 1 for every pair of vertices u and v in H. In Section 2.1, we prove that every 2-connected, 2-heavy and almost distance-hereditary graph is Hamiltonian, where a graph G is 2-heavy if the degrees of at least two end vertices of |V (G)| each claw in G are greater than or equal to 2 . In Section 2.2, we assume that G is a connected claw-free graph satisfying the following condition: For every maximal induced path P of length p ≥ 2 with end vertices u and v holds d(u) + d(v) ≥ |V (G)| − p + 2. Under this assumption, we prove that G has a 2-dominating induced cycle, and then G is Hamiltonian. In Section 2.3, we consider the graph G satisfying this condition: For each vertex u in |V (G)| G it holds: max{d(x) | d(x, u) = 2 and x = u} ≥ 2 . It is proved that G is Hamiltonian if G is 3-connected, claw-free and hourglass-free. In 1997, J. Bang-Jensen and G. Gutin conjectured that an edge-colored complete graph G has a properly colored Hamiltonian path if and only if G has a spanning subgraph consisting of a properly colored path C0 and a (possibly empty) collection of properly colored cycles C1,C2,...,Cd such that V (Ci) ∩ V (Cj) = ∅ provided 0 ≤ i < j ≤ d. In Chapter 3, we prove this conjecture. i ii Another interesting topic in graph theory is the research on the existence of a 2-factor in a graph. In Chapter 4, we consider a special 2-factor, the complementary cycles in jump graphs and we give a characterization for jump graphs containing complementary cycles. In the second part of this thesis, we consider tournaments. T. Yao, Y. Guo and K. Zhang (Discrete Appl. Math. 99 (2000), 245 – 249) proved that every strong tour- nament contains a vertex x such that each arc going out of x is pancyclic. Moreover, they conjectured that each k-strong tournament contains k vertices whose out-arcs are pancyclic. A. Yeo (J. Graph Theory 50 (2005), 212 – 219) proved that each 3-strong tournament contains two vertices x, y such that all out-arcs of x and y are pancyclic. In addition, A. Yeo gave an infinite class of k-strong tournaments for all k ∈ IN, each of which contains at most 3 vertices whose out-arcs are pancyclic, and gave a conjecture that each 2-strong tournament contains 3 vertices whose out-arcs are pancyclic. In Section 6.1, we extend the result of Yeo mentioned above and prove that each 2- strong tournament contains two vertices v1 and v2 such that every out-arc of vi is pancyclic for i = 1, 2. Our proof yields a polynomial algorithm to find such two vertices. In Section 6.2, we prove that each 3-strong tournament contains at least 3 vertices whose out-arcs are pancyclic. That implies that the conjecture of Yao, Guo and Zhang is true for the 3-strong tournament but the conjecture of Yeo is still open only for exactly 2-strong tournaments. Because there are k-strong tournaments for all k ∈ IN containing at most 3 vertices whose out-arcs are pancyclic, it gives rise to an interesting problem: How many vertices does a tournament contain such that all out-arcs of those vertices are 4-pancyclic? In Section 6.3, we show that each s-strong tournament with s ≥ 2 contains at least s + 1 vertices whose out-arcs are 4-pancyclic. The completion of this dissertation was made possible through the support and co- operation of many individuals. I am grateful for my advisor, Prof. Dr. Yubao Guo, who introduced me to graph theory and provided thoughtful guidance and encouragement. I would like to thank Prof. Dr. Eberhard Triesch for his support as co-referee. I also ex- tend special thanks to Prof. Dr. Dr. h.c. Hubertus Th. Jongen for the generous support during the last few years. Thanks also to Dipl.-Math. Hans-Erik Giesen, PD Dr. Harald G¨unzel,Prof. Dr. Gregory Gutin, Dr. Guido Helden, M.Sc. Wei Meng, M.Sc. Ruijuan Li, Prof. Dr. Shengjia Li and Mrs. Roswitha Gillessen for the excellent cooperation and help- ful advices. Moreover, I would like to thank Prof. Dr. Gerhard Hiß for admitting me as an associate member of “Graduiertenkolleg: Hierarchie und Symmetrie in mathematischen Modellen (DFG), RWTH Aachen University, Germany”. Finally, I owe my greatest debts to my family. Particularly, I would like to thank my wife, Ping Zhang, and our parents for their sacrifices and support. Special thanks to my daughter, Xiaoyue, who reminds me daily that miracles exist everywhere around us. Contents Preface i I Some Hamiltonian Graphs and Related Topics 1 1 Introduction 3 1.1 Terminology and Notation . 3 1.2 Some Well-known Results on the Hamiltonian Problem . 5 1.2.1 Four Classical Results on Hamiltonian Graphs . 5 1.2.2 Some Generalizations . 6 1.2.3 Forbidden Subgraphs and Closure Concept . 7 2 Hamiltonian Graphs 11 2.1 Almost Distance-hereditary Graphs . 11 2.2 Graphs with Ore-type Condition . 17 2.3 Graphs with Fan-type Condition . 24 3 Properly Colored Hamiltonian Paths 29 3.1 PCHP Conjecture . 29 3.2 The Proof for PCHP Conjecture . 30 4 Complementary Cycles in Jump Graphs 41 4.1 Some Results on Jump Graphs . 41 4.2 Complementary Cycles in Jump Graphs . 43 II Out-arc Pancyclic Vertices in Tournaments 55 5 Introduction 57 5.1 Terminology and Notation . 57 5.2 Some Results on Tournaments . 58 6 Out-arc Pancyclicity of Vertices in Tournaments 63 6.1 Out-arc Pancyclic Vertices in 2-strong Tournaments . 63 6.2 Out-arc Pancyclic Vertices in 3-strong Tournaments . 66 6.3 Out-arc 4-pancyclic Vertices . 71 Bibliography 107 Index 113 iii Part I Some Hamiltonian Graphs and Related Topics 1 Chapter 1 Introduction In this thesis, we only consider finite graphs without loops and multiple edges. At this point, we introduce some definitions and notations as well as some well-known results of Hamiltonian graphs. For those not defined here we refer to [21]. The definitions and notations of digraphs will be introduced in Chapter 5. 1.1 Terminology and Notation Let G = (V (G),E(G)) be a graph.
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