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Coloring of Signed Graphs UNIVERSITAT¨ PADERBORN Fakult¨atf¨urElektrotechnik, Informatik und Mathematik Dissertation Coloring of Signed Graphs by Yingli Kang Adviser: Prof. Dr. Eckhard Steffen in partial fulfillment of the requirements for the degree of doctor rerum naturalium (Dr. rer. nat.) Paderborn 2017 ii Abstract The study on signed graphs have been one of the hot research fields in the past few years. Theories on ordinary graphs have been generalized to signed graphs in many major aspects, such as the areas of flows, circuit covers, ho- momorphisms and so on. Graph colorings theory, which is strongly related to these aspects, has a central position in discrete mathematics. However, there are very few knowledges known on colorings of signed graphs so far. The thesis is devoted to generalize a series of concepts, results and methods on vertex colorings of graphs to signed graphs for the first time. In particular, we introduce the notions of circular colorings and related integer colorings and list colorings for signed graphs. Some fundamental results for each notion are proved. Analogues of some classical results like Brooks' Theorem and Haj´os' Theorem for signed graphs are presented. Moreover, the relation between a signed graph and its underlying unsigned graph is investigated, especially for chromatic numbers and list-chromatic numbers. Some exclusive features for signed graphs such as the chromatic spectrum are studied. The thesis concludes with a result on 3-coloring of unsigned planar graph. Zusammenfassung Signierte Graphen sind ein hochinteressantes und aktives Forschungsgebiet der Graphentheorie mit vielf¨altigenAnwendungen in anderen Disziplinen wie z.B. der Physik oder der Soziologie. Viele graphentheoretische Konzepte, wie z.B. Fl¨usseoder k¨urzesteKreis¨uberdeckungen, wurden auf signierte Graphen verallgemeinert. Unter diesem Aspekt sind F¨arbungensignierter Graphen von besonderem mathematischen Interesse, da viele Konzepte, die f¨urunsignierte Graphen ¨aquivalent sind, dies f¨ursignierte Graphen nicht sind. In dieser Arbeit werden vornehmlich Eckenf¨arbungen auf signierten Graphen studiert. Es wird das Konzept der zirkul¨arenF¨arbungvon sig- nierte Graphen eingef¨uhrtund die darauf basierenden Parameter wie z.B. die iii zirkul¨arechromatische Zahl, die chromatische Zahl und die listenchromatische Zahl werden studiert. Klassische Ergebnisse der Graphnetheorie, wie die S¨atzevon Brooks und Haj´oswerden auf signierte Graphen verallgemeinert. Das chromatische Spek- trum signierter Graphen wird bestimmt. Die Beziehung zwischen der chro- matischen Zahl des signierten und der chromatischen Zahl des unterliegen- den unsignierten Graphen studiert. Weiterhin werden die unterschiedlichen F¨arbungskonzepte verglichen. Die Arbeit schließt mit einem Ergebnis zu un- signierten 3-f¨arbbarenplanaren Graphen. iv Acknowledgements First and foremost, I would like to thank my advisor Prof. Dr. Eckhard Steffen for the continuous support of my Ph.D. study and research. It has been an honor to become one of his Ph.D. student. He has encouraged me, both consciously and patiently, to try some other new methods besides dis- charging. I appreciate all his contributions of time, ideas to make my Ph.D. experience memorable and stimulating. I am also thankful for his support and understanding during my pregnancy. The joy, enthusiasm and rigorous logical thinking he has for his research was contagious and motivational for me, even during tough times to finish my Ph.D. thesis. I would like to express my sincere gratitude to Prof. Dr. Xuding Zhu for recommending me to be a doctor candidate in the research group of Prof. Dr. Eckhard Steffen. Although he is a famous Mathematician, he is still as common as an old shoe. Thanks to his useful suggestions on doing research and how to give a good talk. My sincere thanks also go to Prof. Dr. Yingqian Wang who always encourages me to do Ph.D. study since the first year of my postgraduate. Thanks to my colleague Michael Schubert who is a very good friend to me since he first time came to China. His help both academically and in daily life make me familiar with the totally different culture and the new country. Talking with him always not only for all their useful suggestions but also for being there to listen when I needed an ear. Thanks to Astrid Canisius who made all administrative issues during my stay at University Paderborn very easy. Lastly, I would like to thank my family: my parents Haixia Xia and Ji'an Kang, and my young brother Liuyong Kang, for all their love and encour- agement and for supporting me spiritually throughout my life. To my lovely son Lois Yuanhao Jin who accompany me and bring joy to me (bittersweet moment). And most of all for my loving, patient, supportive and encouraging husband Ligang Jin who is also my college. His faithful support during the final stages of this Ph.D. is so appreciated. v The research in the thesis is supported by the Fellow of the International Graduate School "Dynamic Intelligent Systems". vi To my parents, to Ligang and our son Lois. vii viii Contents Abstract . iii Acknowledgements . v List of Figures . xiii 1 Introduction 1 1.1 Signed graphs, balances and switchings . 1 1.2 Colorings of signed graphs . 3 2 Circular chromatic number χc of signed graphs 11 2.1 (k; d)-colorings of a signed graph . 12 2.1.1 A recoloring technique: Updating . 13 2.1.2 Basic properties of (k; d)-colorings . 16 2.1.3 χc((G; σ)): from infimum to minimum . 24 2.2 Circular r-colorings of a signed graph . 24 2.3 Relation between χc((G; σ)) and χ((G; σ)) . 28 2.3.1 Relations in general . 28 2.3.2 Signed graphs for which χc = χ − 1 . 29 2.3.3 Improved lower bound . 32 2.4 Signed graphs for which χc = χ . 33 2.4.1 Interval colorings of signed graphs . 34 2.4.2 Construction . 35 3 Chromatic number χ of signed graphs 37 3.1 Some basic properties . 37 3.2 Chromatic spectrum of a graph . 40 ix 3.2.1 Determination of mχ(G).................. 40 3.2.2 Chromatic critical signed graphs . 41 3.2.3 Proof of Theorem 3.6 . 42 3.3 An analogue of Brooks' Theorem for signed graphs . 43 3.4 First Haj´os-like theorem for signed graphs . 47 3.4.1 Introduction . 47 3.4.2 Signed bi-graphs . 49 3.4.3 Graph operations on signed bi-graphs . 49 3.4.4 Useful lemmas . 51 3.4.5 Proof of the theorem . 54 4 Signed chromatic number χ± of signed graphs 59 4.1 Preliminary . 59 4.2 Signed chromatic spectrum of a graph . 60 4.2.1 Signed chromatic critical graph . 61 4.2.2 Proof of Theorem 4.4 . 62 4.3 Second Haj´os-like theorem for signed graphs . 62 4.4 Relation between χ and χ± .................... 65 5 Choosability in signed graphs 67 5.1 Definitions and basic properties . 67 5.2 Choosability in signed planar graphs . 69 5.2.1 5-choosability . 70 5.2.2 4-choosability . 73 5.2.3 3-choosability . 79 6 3-colorings of planar graphs 85 6.1 Introduction . 85 6.2 Notations and formulation of the main theorem . 88 6.3 Proof of the main theorem . 91 6.3.1 Structural properties of the minimal counterexample G 91 6.3.2 Discharging in G ...................... 99 x 7 Conclusion and future work 103 Bibliography 105 xi xii List of Figures 2.1 A signed graph (G; σ) with χc((G; σ)) = χ((G; σ)) − 1. 29 ∗ ∗ 2.2 χ((K4 ; σ4)) − 1 = χc((K4 ; σ4)) = 4. 31 ∗ ∗ ∗ ∗ 2.3 χ((Q4; σ4)) − 1 = χc((Q4; σ4)) = 5. 32 3.1 A construction of (G3; σ3). 39 3.2 Operation (3) . 47 3.3 Operation (sb3)........................... 50 4.1 Two signed graphs with jχ − χ±j = 1 . 66 5.1 The graph G3 ............................ 72 5.2 The graph H ............................ 72 5.3 graph T ............................... 83 6.1 Two graphs as counterexamples to the statement . 87 6.2 Chord, claw, biclaw, triclaw and combclaw of a cycle . 89 xiii xiv Chapter 1 Introduction 1.1 Signed graphs, balances and switchings The notion of signed graphs was first introduced by Harary [21] in 1953, as a mathematical model for certain problems in social psychology. The social structure of a group of persons is often represented by a graph, for which each vertex represents a person and two vertices are connected by an edge (i.e., adjacent) if and only if they know each other. The motivation for defining signed graphs arise from the fact that psychologists sometimes describe the relation between two persons as liking, disliking or indifference. A signed graph is a graph together with a sign of \ + " or \ − " on each edge, representing the emotion of liking or disliking. Hence, the notion of signed graphs is a generalization of ordinary graphs. The standard definitions used in the theory of graphs may be found in [54]. We consider a graph to be finite and simple, i.e., with no loops or multiple edges. The vertex set of a graph G is denoted by V (G), and the edge set by E(G). Let G be a graph and σ : E(G) ! f1; −1g be a mapping. The pair (G; σ) is called a signed graph. We say that G is the underlying graph of (G; σ) and σ is a signature of G. The sign of an edge e is the value σ(e). An edge is positive if it has a positive sign; otherwise, the edge is negative. The set Nσ = fe: σ(e) = −1g is the set of negative edges of (G; σ) and E(G)−Nσ the 1 2 Chapter 1 Introduction set of positive edges.
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