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Hamiltonian Properties of Generalized Halin Graphs Hamiltonian Properties of Generalized Halin Graphs Name : Ahmad Mahmood Qureshi Session : 2003-2008 Registration No. : 01-GCU-Ph.D.-SMS-2003 Abdus Salam School of Mathematical Sciences GC University Lahore, Pakistan Hamiltonian Properties of Generalized Halin Graphs Submitted to Abdus Salam School of Mathematical Sciences GC University Lahore, Pakistan In the partial fulfillment of the requirements for the award of degree of Doctor of Philosophy in Mathematics By Name : Ahmad Mahmood Qureshi Session : 2003-2008 Registration No. : 01-GCU-Ph.D.-SMS-2003 Abdus Salam School of Mathematical Sciences GC University Lahore, Pakistan ii DECLARATION I, Mr. Ahmad Mahmood Qureshi Registration No. 01-GCU-Ph.D.-SMS-2003 student at School of Mathematical Sciences GC University Lahore in the subject of Mathematics session ( 2003-2008), hereby declare that the matter printed in thesis titled “Hamiltonian Properties of Generalized Halin Graphs” is my own work and that (i) I am not registered for similar degree elsewhere contemporaneously. (ii) No major work had already been done by me or anybody else on the topic; I worked on for the Ph.D degree. (iii) The work I am submitting for the Ph.D degree has not already been submitted elsewhere and shall not in future be submitted by me for obtaining similar degree from any other institution. Dated : ------------------------- ------------------------------------ Signature of Deponent iii RESEARCH COMPLETION CERTIFICATE Certified that the research work contained in this thesis titled “Hamiltonian Properties of Generalized Halin Graphs” has been carried out and completed by Mr. Ahmad Mahmood Qureshi Registration No. 01-GCU-Ph.D.-SMS-2003 under my supervision. ----------------------------- ------------------------------- Date Supervisor Submitted Through Prof . Dr. A. D. Raza Choudary -------------------------------- Director General Controller of Examination Abdus Salam School of Mathematical Sciences GC University Lahore, GC University Lahore, Pakistan. Pakistan. iv Table of Contents Table of Contents v Abstract vii Acknowledgements ix 1 Introduction 1 1.1 Basic terminology . 1 1.2 Isomorphic graphs and subgraphs . 2 1.3 Connected graphs and connectivity . 4 1.4 Common classes of graphs . 6 1.5 Planar graphs . 8 2 Hamiltonian Graphs 11 2.1 Historical note . 11 2.2 Some de¯nitions . 13 2.3 Hamiltonian planar graphs . 14 3 Halin graphs 17 3.1 Introduction . 17 3.2 Structural properties . 18 3.3 Hamiltonian properties . 20 3.4 Two variants of Halin graphs . 22 4 3-Halin graphs 25 4.1 k-Halin graphs . 25 4.2 Hamiltonicity of 3-Halin graphs . 26 4.3 Pancyclicity of 3-Halin graphs . 30 v 5 15-Halin graphs 34 5.1 14-Halin graphs with connected core . 34 5.2 Almost k-Halin graphs . 37 5.3 14-Halin graphs . 46 5.4 15-Halin graphs . 48 6 Conclusion 51 Bibliography 53 vi Abstract A Halin graph is a graph H = T [ C, where T is a tree with no vertex of degree two, and C is a cycle connecting the end-vertices of T in the cyclic order determined by a plane embedding of T . Halin graphs were introduced by R. Halin [16] as a class of minimally 3-connected planar graphs. They also possess interesting Hamiltonian properties. They are 1-Hamiltonian, i.e., they are Hamiltonian and remain so after the removal of any single vertex, as Bondy showed (see [23]). Moreover, Barefoot proved that they are Hamiltonian connected, i.e., they admit a Hamiltonian path be- tween every pair of vertices [1]. Bondy and Lov¶asz[6] and, independently, Skowronska [33] proved that Halin graphs on n vertices are almost pancyclic, more precisely they contain cycles of all lengths l (3 · l · n) except possibly for a single even length. Also, they showed that Halin graphs on n vertices whose vertices of degree 3 are all on the outer cycle C are pancyclic, i.e., they must contain cycles of all lengths from 3 to n. In this thesis, we de¯ne classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties. In chapter 4, we de¯ne k-Halin graph in the following way. A 2-connected planar graph G without vertices of degree 2, possessing a cycle C such that (i) all vertices of C have degree 3 in G, and (ii) G ¡ C is connected and has at most k cycles is called a k-Halin graph. vii viii A 0-Halin graph, thus, is a usual Halin graph. Moreover, the class of k-Halin graphs is contained in the class of (k + 1)-Halin graphs (k ¸ 0). We shall see that, the Hamiltonicity of k-Halin graphs steadily decreases as k increases. Indeed, a 1-Halin graph is still Hamiltonian, but not Hamiltonian con- nected, a 2-Halin graph is not necessarily Hamiltonian but still traceable, while a 3-Halin graph is not even necessarily traceable. The property of being 1-Hamiltonian, Hamiltonian connected or almost pancyclic is not preserved, even by 1-Halin graphs. However, Bondy and Lov¶asz'result about the pancyclicity of Halin graphs with no inner vertex of degree 3 remains true even for 3-Halin graphs. The property of being Hamiltonian persists, however, for large values of k in cubic 3-connected k-Halin graphs. In chapter 5, it will be shown that every cubic 3- connected 14-Halin graph is Hamiltonian. A variant of the famous example of Tutte [37] from 1946 which ¯rst demonstrated that cubic 3-connected planar graphs may not be Hamiltonian, is a 21-Halin graphs. The cubic 3-connected planar non-Hamiltonian graph of Lederberg [21], Bos¶ak[7] and Barnette, which has smallest order, is 53-Halin. The sharpness of our result is proved by showing that there exist non-Hamiltonian cubic 3-connected 15-Halin graphs. Acknowledgements I am grateful to Almighty ALLAH, most Gracious and Merciful, for enabling me to complete this research thesis. I wish to thank Professor Dr. Tudor Zam¯rescu, my supervisor, for his many valuable suggestions, constant support and guidance. He provided me the opportunity to stay at the University of Dortmund, Germany during the ¯nal span of my work and utilize the resources there. I extend special thanks to Professor Dr. A. D. R. Choudary, director general, Abdus Salam School of Mathematical Sciences (ASSMS) for his support and encour- agement throughout my candidature. My thanks go to all the foreign faculty at ASSMS for giving us their precious time to work with us and to improve our skills in Mathematics. I am grateful to my colleagues at ASSMS for maintaining a friendly and collabo- rative work environment and spending together all good and rough times. Especially, I wish to thank Ms. Shabnam Malik for precious discussions on the subject. Last, but most importantly, I would like to thank my family, in particular my wife, without whose constant support, I would never had succeeded in completing this thesis. Of course, the prayers of my parents has made my e®orts fruitful. ASSMS, Lahore A. M. Qureshi February, 2008. ix Chapter 1 Introduction We begin our study by introducing some of the basic concepts that we shall encounter throughout our investigation. 1.1 Basic terminology A graph G is a ¯nite non-empty set of objects called vertices together with a (possibly empty) set of unordered pairs of distinct vertices of G called edges. The vertex set of G is denoted by V (G), while the edge set is denoted by E(G). The edge e = fu; vg is said to join the vertices u and v. If e = fu; vg is an edge of a graph G, then u and v are adjacent vertices. Edge e is incident with each of the two vertices u and v. Furthermore, if e1 and e2 are distinct edges of G incident with a common vertex, then e1 and e2 are adjacent edges. It is convenient to henceforth denote an edge by uv or vu rather than fu; vg. The cardinality of the vertex set of a graph G is called the order of G and is commonly denoted by n(G) or more simply by n when the graph under consideration is clear; the cardinality of its edge set is the size of G and is often denoted by m(G) 1 2 or m. Since the vertex set of every graph is nonempty, the order of every graph is at least 1. A graph with exactly one vertex is called a trivial graph. The degree of a vertex v in a graph G is the number of edges incident with v and is denoted by dGv or simply by d(v). This is also the number of vertices in G that are adjacent to v. Two adjacent vertices are referred to as neighbours of each other. The set N(v) of neighbours of a vertex v is called the neighbourhood of v. Thus d(v) = cardN(v). A vertex of even degree is called an even vertex, while a vertex of odd degree is an odd vertex. A vertex of degree 0 is referred to as an isolated vertex, and a vertex of degree 1 is a leaf. The minimum degree of G is the minimum degree among the vertices of G and is denoted by ±(G). The maximum degree is de¯ned similarly and is denoted by ¢(G). So, if G is a graph of order n and v is any vertex of G, then 0 · ±(G) · d(v) · ¢(G) · n ¡ 1: Every edge in a graph is incident with two vertices, hence, when the degrees of all vertices are summed, each edge is counted twice. Theorem 1.1.1. If G is a graph of size m, then X d(v) = 2m: v2V (G) Corollary 1.1.2.
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