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

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

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 deﬁnitions ...... 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 between 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 deﬁne classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties.

In chapter 4, we deﬁne 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 connected, 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 ﬁrst 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 Zamfirescu, 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 final 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 efforts 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 ﬁnite 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 = {u, v} is said to join the vertices u and v. If e = {u, v} 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 {u, v}.

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 δ(G). The maximum degree is deﬁned similarly and is denoted by ∆(G). So, if G is a graph of order n and v is any vertex of G, then

0 ≤ δ(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. v∈V (G) Corollary 1.1.2. In any graph, there is an even number of odd vertices.

1.2 Isomorphic graphs and subgraphs

A graph G1 is said to be isomorphic to a graph G2 if there exists a bijective mapping ϕ : V (G1) → V (G2) such that ϕ preserves adjacency and non-adjacency. 3

That is uv ∈ E(G1) if and only if ϕ(u)ϕ(v) ∈ E(G2). In this case ϕ is called an isomorphism from G1 to G2. It is easy to see that isomorphy is an equivalence ∼ relation on graphs. If G1 is isomorphic to G2, then we denote this by writing G1 = G2. A graph H is called a subgraph of a graph G, written H ⊂ G, if V (H) ⊂ V (G) and E(H) ⊂ E(G). In such a case, we also say that G is a supergraph of H. If a subgraph H of G has the same order as G, then H is called a spanning subgraph of

G. A subgraph H of a graph G is called an induced subgraph of G if whenever u and v are vertices of H and uv is an edge of G, then uv is an edge of H as well.

Any subgraph of a graph G can be obtained by removing vertices and edges from

G. If v ∈ V (G) and cardV (G) ≥ 2, then G − v denotes the subgraph with vertex set V (G) − {v} and whose edges are all those of G not incident with v. If e ∈ E(G), then G − e is the subgraph having vertex set V (G) and edge set E(G) − {e}. The deletion of a set of vertices or set of edges is deﬁned analogously. These concepts are illustrated in Figure 1.1.

Figure 1.1: 4

1.3 Connected graphs and connectivity

Let u and v be vertices of a graph G.A u − v path in G is a subgraph P ⊂ G with V (P ) = {u = u0, u1, u2, ..., uk−1, uk = v} and E(P ) = {ui−1ui : 1 ≤ i ≤ k}. We emphasize that all vertices of a path are distinct. The vertices u and v are called endpoints of P . The number k is the length of P , and coincides with m(P ). A path of order n is denoted by Pn. Two u − v paths in G are internally disjoint if they do not share any vertex except u and v.

Identifying u and v in the deﬁnition above yields the concept of a cycle. An acyclic graph has, by deﬁnition, no cycles. A k-cycle is a cycle of order k, denoted by Ck. A 3-cycle is also referred to as a triangle. A cycle of odd length is an odd cycle; otherwise it is an even cycle.

We now introduce the concept of connected and disconnected graphs. A vertex u is said to be connected to a vertex v in a graph G if there exists a u − v path in G.

A graph G is connected if every two vertices of G are connected. A graph G that is not connected is called disconnected.

The relation “ is connected to” is an equivalence relation on the vertex set of every graph G. Each subgraph induced by the vertices in a resulting equivalence class is called a connected component or simply a component of G. Equivalently, a component of a graph G is a connected subgraph of G not contained in any other connected subgraph of G.

Connected graphs can be made disconnected by removing vertices or edges. A vertex v in a connected graph G is a cut-vertex of G if G − v is disconnected. Anal- ogously, an edge e of a connected graph G is a bridge of G if G − e is disconnected. 5

Of course, the vertices incident to a bridge are likely to become cut-vertices.

The idea of connectivity of a graph extends the concepts of cut-vertex and bridge.

Two invariants called connectivity and edge connectivity are useful in deciding which of two graphs is ‘more connected’.

The vertex connectivity or simply the connectivity κ(G) of a graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph.

A graph G is said to be k-connected, k ≥ 1, if κ(G) ≥ k. Thus G is 1-connected if and only if G is non-trivial and connected. In general, G is k-connected if and only if the removal of fewer than k vertices results in neither a disconnected nor a trivial graph. A useful characterization of k-connected graphs given by Whitney, is the following.

Theorem 1.3.1. A graph G of order n ≥ 2 is k-connected (1 ≤ k ≤ n − 1) if and only if for each pair u, v of distinct vertices there are at least k internally disjoint u − v paths in G.

Connectivity has an edge analogue. The edge connectivity λ(G) of a graph G is the minimum number of edges whose removal from G results in a disconnected or trivial graph. Thus λ(G) = 1 if and only if G is connected and contains a bridge.

A graph G is k-edge connected, k ≥ 1, if λ(G) ≥ k. Equivalently, G is k-edge connected if the removal of fewer than k edges from G results in neither a disconnected graph nor a trivial graph. Theorem 1.3.1 has an edge analog proved by Ford and

Fulkerson.

Theorem 1.3.2. A graph G is k-edge connected if and only if for every two distinct vertices u and v of G there exist at least k edge disjoint u − v paths in G. 6

Connectivity, edge connectivity and minimum degree are related by an inequality due to Whitney.

Theorem 1.3.3. For any graph G,

κ(G) ≤ λ(G) ≤ δ(G).

Both inequalities in theorem 1.3.3 can be strict as the graph in Figure 1.2 shows.

Figure 1.2: Graph G with κ(G) = 1, λ(G) = 2 and δ(G) = 3

A graph G is called cubic if δ(G) = ∆(G) = 3.

Theorem 1.3.4. If G is a cubic graph, then κ(G) = λ(G).

1.4 Common classes of graphs

A graph G is regular of degree r if d(v) = r for each vertex v of G. Such graphs are called r-regular. A 3-regular graph is also called cubic.

A graph G is complete if every two distinct vertices of G are adjacent. A complete graph of order n, denoted by Kn is a regular graph of degree n − 1. Moreover Kn has the maximum possible size among all graphs of order n. Since every two distinct 7

¡n¢ n(n−1) vertices of Kn are joined by an edge, the size of Kn is 2 = 2 .

A graph G is a bipartite graph if V (G) can be partitioned into two sets V1 and V2 in such a way that no two vertices from the same set are adjacent. The sets V1 and

V2 are called partite sets or the colour classes of G and (V1,V2) is a bipartition of G. In fact, a graph being bipartite means that the vertices of G can be coloured with two colours, so that no two adjacent vertices have the same colour. We shall depict bipartite graphs with their vertices coloured black and white to show one possible bipartition.

It is not always easy to tell at a glance whether a graph is bipartite. We mention one of the most widely used characterizations of bipartite graphs, which was obtained by K¨onig(1916).

Theorem 1.4.1. A graph is bipartite if and only if it has no odd cycle.

A complete bipartite graph is the bipartite graph with bipartition (V1,V2) with cardV1 = p and cardV2 = q such that each vertex of V1 is adjacent to each vertex of

V2. It is denoted by Kp,q. In particular K1,q is called a star. A special class of bipartite graphs are the trees. A tree is an acyclic connected graph. The bipartite nature of a tree follows from the absence of cycles (in particular odd cycles). The paths Pn and stars K1,q are trees. There are a number of ways to characterize trees. The following theorem lists several simple ones.

Theorem 1.4.2. Let T be a graph on n vertices. Then the following statements are equivalent.

(i) T is a tree; 8

(ii) T contains no cycle, and has n − 1 edges;

(iii) T is connected, and has n − 1 edges;

(iv) T is connected, and every edge is a bridge;

(v) any two vertices of T are connected by exactly one path;

(vi) T contains no cycles, but the addition of any new edge creates exactly one cycle.

A basic property of trees that is useful when using mathematical induction to prove theorems involving trees is the following.

Theorem 1.4.3. Every non-trivial tree has at least two leaves.

1.5 Planar graphs

Intuitively, a graph G is called a planar graph if it can be drawn in the plane so that no two edges intersect except at a vertex. Such a drawing is a plane embedding of G. Thus, a plane embedding of a graph G is a function ϕ that assigns to each vertex a point in the plane R2 and to each edge an arc in R2 in such a way that the arcs ϕ(e1) and ϕ(e2) meet at a point p if and only if p = ϕ(v) for some vertex v which is incident in G to the edges e1 and e2. If the graph G admits a plane embedding, it is called planar.A plane graph is a particular plane embedding of a planar graph. It S is usually represented by all images of ϕ. Let setG = ϕ(e). This set depends e∈E(G) on ϕ.

A plane graph G yields a set setG ⊂ R2. The components of R2 − setG are called faces. There is always exactly one face of G that is unbounded, called inﬁnite face.

A plane graph G can always be re-embedded in the plane so that a given face of G becomes the inﬁnite face. Hence a planar graph G can always be realized in the plane 9

Figure 1.3: A planar graph and its plane embedding so that any vertex or edge lies on the boundary of its inﬁnite face.

Euler’s formula for planar graphs (given by Leonhard Euler in 1758) relates the order, size and number of faces of plane graphs.

Theorem 1.5.1. Suppose a connected plane graph of order n and size m has f faces.

Then

n − m + f = 2.

By Euler’s formula, all planar embeddings of a connected graph G have the same number of faces. Euler’s formula has many applications, particularly for plane graphs all faces of which are bounded by cycles of length at least 3.

Corollary 1.5.2. If G is a planar graph of order n ≥ 3 and size m, then

m ≤ 3n − 6.

If G is triangle-free, then m ≤ 2n − 4.

Corollary 1.5.2 provides a necessary condition for a graph to be planar, and so its negation is a suﬃcient condition for a graph to be non-planar. Non-planarity of K5 and K3,3 follows immediately from Theorem 1.5.2. For K5, we have m = 10 > 9 =

3n − 6. Since K3,3 is triangle-free, we have m = 9 > 8 = 2n − 4. K5 and K3,3 are the 10

crucial graphs that led to a characterization of a planar graph known as Kuratowski’s theorem.

Two graphs G1 and G2 are homeomorphic if they can be obtained from the same graph G by replacing the edges of G by paths. It is clear that planarity is invariant under homeomorphic transformations of a graph. Therefore, if G is a graph that contains a homeomorph of K5 or K3,3 as a subgraph, then G is non-planar. Kuratowski’s theorem states that the converse of this statement is true as well.

Theorem 1.5.3. A graph G is planar if and only if G does not contain a homeomorph of K5 or K3,3 as a subgraph.

Another characterization of planar graphs is in terms of forbidden minors. Given an edge uv of a graph G, the contraction of edge uv is obtained by identifying the vertices u and v. A graph H obtained by a sequence of edge-contractions is said to be a contraction of G. A graph H is called a minor of a graph G if H can be obtained from G by a succession of edge contractions, edge deletions, and vertex deletions. In

1937, Wagner proved the following analog of Kuratowski’s theorem.

Theorem 1.5.4. A graph is planar if and only if it contains neither K5 nor K3,3 as a minor. Chapter 2

Hamiltonian Graphs

We now embark upon a study of related literature in this chapter. We begin with a historical note on Eulerian and Hamiltonian graphs, and then present necessary terminnology. We conclude the chapter with a study of plane Hamiltonian graphs.

2.1 Historical note

Graph theory takes its origin from challenging puzzles, testing the ingenuity and skill of a challenger. Among such puzzles is the Königsberg bridges problem that asks for a route which would cross each of the seven bridges of Königsberg just once. Euler was the first mathematician who formulated and solved the problem treating it in pure mathematical terms. In 1735 he presented a paper [12] to the

St. Petersburg Academy of Sciences, proving that the solution to the problem is impossible. Euler further generalized his method to such problems comprising of any number and arrangement of islands and bridges. It is unfortunate that Euler’s work remained unknown for around 150 years. In [14] Biggs, Lloyd and Wilson wrote:

“Indeed, the problem was not well known until the end of the 19th century, when

Lucas [24] and Rouse Ball [31] included it in their books on recreational mathematics,

11 12

although a French translation of Euler’s paper had been published earlier by Coupy

[10]”.

I further quote from [14]: “A recreational puzzle related to the Königsberg bridges problem is that of finding the smallest number of pen-stokes needed to draw a given diagram with no line repeated. Poinsot [29] showed that a single stroke is sufficient for an odd number of points all joined in pairs (the complete graph Kn, n odd), but not for an even number of points”. “The term Eulerian graph for a graph which can be drawn with a single pen-stroke is due to König,and appeared in his poineering book [20]”.

An apparently similar but older problem is the Knight’s tour problem which asks for a sequence of moves that takes the Knight to each square on a standard 8 × 8 chessboard exactly once and returning to the starting square. The ﬁrst solution of this problem was discovered in 840 A.D. by the famous chess player al-Adli of Bagdad

[26]. Many other solutions were discovered later but the ﬁrst systematic analysis of the problem was carried out by Euler in 1759 in a long paper [13]. Let G be the graph on 64 vertices associated to the chessboard and the knight moves. In modern graph- theoretic terms, Euler puts the problem of its traceability, then of its homogeneous traceability, and ﬁnally of its hamiltonicity. He discovers techniques yielding long paths in G and techniques rendering these long paths hamiltonian. He also considers the problem on more general n × n boards and proves that on a 5 × 5 board there is no solution to the problem. Advancing further, he considers the corresponding problem on rectangular boards and boards that are cross-shaped. The problem was again taken up in 1771 by Vandermonde [39], who also gave it a systematic treatment 13

and used it to demonstrate his ideas on the geometry of position.

“The ﬁrst general discussion of vertex traversal problems was given by Kirkman

[19] in 1855, who asked which polyhedra allow a cycle passing through each vertex, and described a general class of polyhedra for which no such cycles exists” [14].

In 1857, Hamilton discovered a system of non-commutative algebra and named it

‘The Icosian Calculus’ which can be interpertated in terms of paths on the graph of a regular dodecahedron. He used this graph theoretical interpretation as the basis for a puzzle, ‘The Icosian Game’, the object of which was to ﬁnd paths in the dodeca- hedral graph visiting each vertex. From this game of Hamilton was born the term of

Hamiltonicity [4, 14].

2.2 Some deﬁnitions

A cycle in a graph G that contains every vertex of G is called a Hamiltonian cycle of

G. Thus, a Hamiltonian cycle of G is a spanning cycle of G.A Hamiltonian graph is a graph that contains a Hamiltonian cycle. For n ≥ 3, the graphs Cn and Kn are Hamiltonian. A path in a graph G that contains every vertex of G is called a

Hamiltonian path in G. A graph that contains a Hamiltonian path is called traceable.

Clearly every Hamiltonian graph is traceable but the converse is not true, as the example of a path shows. A graph that admits a Hamiltonian path between every pair of vertices is called Hamiltonian connected. Obviously every Hamiltonian connected graph is Hamiltonian but the converse is not true. Figure 2.1 shows graphs which are non-traceable, traceable (but not Hamiltonian), Hamiltonian (but not Hamiltonian connected) and Hamiltonian connected. 14

Figure 2.1:

A Hamiltonian graph G is k-Hamiltonian if G − V 0 is Hamiltonian for any V 0 ⊆

V (G) such that cardV 0 ≤ k. It is clear that every k-Hamiltonian graph has at least k + 3 vertices and the degree of every vertex is at least k + 2. A graph G is k−edge

Hamiltonian if G − E0 is Hamiltonian for any E0 ⊆ E(G) such that cardE0 ≤ k.

A graph G is said to be uniformly Hamiltonian if each edge of G is in some Hamil- tonian cycle and is avoided by another one.

A graph of order n is pancyclic if it has cycles of all lengths l between 3 and n, and is almost pancyclic, if it contains cycles of all lengths from 3 to n except possibly for one single length. An almost pancyclic graph without cycles of length k (3 ≤ k ≤ n) is called k-almost pancyclic.

2.3 Hamiltonian planar graphs

The interest in Hamiltonian planar graphs arises partly due to their connection with the four-color theorem, and partly from a related conjecture of Tait (1880), that every cubic 3-connected planar graph is Hamiltonian (fuller references for the results in this section may be found in Gr¨unbaum [15]). We mention a necessary condition, due to

Grinberg, for a plane graph to be Hamiltonian. 15

Theorem 2.3.1. Let G be a plane graph of order n with Hamiltonian cycle H. If G

0 has fi faces of length i (i-gons) inside H and fi faces of length i outside H, then Xn 0 (i − 2)(fi − fi ) = 0. i=3 Being necessary, Grinberg’s condition can be used to show that graphs are not

Hamiltonian. The argument can often be simpliﬁed using modular arithmetic. In the following theorem, we apply this to the “Tutte triangle”.

Figure 2.2: Tutte triangle T

Theorem 2.3.2. There is no Hamiltonian path between vertices a and b of the Tutte triangle T in Figure 2.2.

Proof. Suppose on the contrary that there exists a Hamiltonian path joining vertices a and b in T . Since vertex c has degree 2, any Hamiltonian path goes through c in a unique way along the path dce. Without destroying the Hamiltonian path we may subdivide edge ce of T by a new vertex x and name the new graph T 0.

Additionally, let us introduce vertices y and z and the path P = ayzb in T 0, so that the new graph G = T 0 ∪ P is Hamiltonian (the assumed Hamiltonian path between a and b in T 0 and the path P gives Hamiltonian cycle H in G (see Figure 2.3).

Now, by Grinberg’s theorem, for the Hamiltonian cycle H in G,

0 0 0 2(f4 − f4) + 3(f5 − f5) + 9(f11 − f11) = 0 16

Figure 2.3:

In mod 3 this reduces to

0 2(f4 − f4) ≡ 0(mod3)

0 Since there is only one 4-gon in G, either f4 = 1 and f4 = 0 (if the 4-gon is inside

0 H), or f4 = 0 and f4 = 1 (if the 4-gon is outside H). Thus, the last congruence becomes −1(±1) ≡ 0(mod3), a contradiction. Chapter 3

Halin graphs

This chapter deals with a class of graphs called Halin graphs. They were introduced by R. Halin in 1971 as an example of minimally 3-connected graphs. We describe here a number of structural and Hamiltonian properties of Halin graphs.

Two closely related variants of Halin graphs are also presented at the end.

3.1 Introduction

Extending the works of Tutte [38], Dirac [11] and Robertson [30], where they study, respectively, 3-connected, 2-connected and 4-connected graphs, Halin [16] studied those n-connected graphs which after the deletion of any edge lose this property, and called them minimally n-connected graphs. As an example of a minimally 3-connected graph, he gave the following construction.

Let a ﬁnite tree T without vertices of degree 2 be drawn in the plane. Then draw in the plane a cycle C which has exactly the leaves of T as its vertices such that the resulting graph H = T ∪ C remains planar. Lov´aszand Plummer [23] named

H a Halin graph. Despite of the simple structure Halin graphs have, they possess interesting structural and Hamiltonian properties. In the following sections we give

17 18

a review of these properties.

3.2 Structural properties

Let H = T ∪ C be a Halin graph. If the tree T is a star, then H is a wheel and is the simplest type of Halin graph. A fan in H is induced by a non leaf vertex v of T , which is adjacent to exactly one other nonleaf of T , and the leaves adjacent to v. If T has more than one non leaves, then H contains at least two fans. Contracting a fan to a single vertex on C, gives rise to another Halin graph. This reduction procedure thus, recursively deﬁnes the class of Halin graphs. Successive application of this reduction procedure to any Halin graph ultimately reduces it to a wheel.

Suppose H has order n. While the size of the tree T remains n − 1, the size of the cycle C may vary within bounds, with the largest value when T is a star. Moreover, for a given length of cycle C, H may have more than one edge bipartitions {T , C} with isomorphic trees. Horton and Parker [18] demonstrated these results.

Theorem 3.2.1. Let H = T ∪ C be a Halin graph of order n, with C of size k. Then n + 1 ≤ k ≤ n − 1. 2

Theorem 3.2.2. Let {Ti, Ci} i = 1, 2, ..., r be the edge bipartitions of a Halin graph

H, where for all indices i and j, Ci and Cj are of the same size. If Ci 6= Cj, then the corresponding trees Ti and Tj are isomorphic.

As a matter of fact the edge bipartitions of a Halin graph into a tree and cycle are very few. An embedding of a planar graph is called H-feasible if the resulting plane graph is a Halin graph. SysÃlo and Proskurowski [36] gave bounds for the number of

H-feasible embeddings of planar graphs. 19

Theorem 3.2.3. A planar graph can have at most 4 diﬀerent H-feasible embeddings.

Corollary 3.2.4. The only plane graph which is a Halin graph in four of its embeddings is K4.

Theorem 3.2.5. The only Halin graph with exactly three H-feasible embeddings is the geometric dual of K5 − e, where e is any edge of K5.

A special class of Halin graphs, called necklaces, is deﬁned as follows.

0 0 Let Ph+2 and Ph be paths with their vertices labelled from 0 to h + 1 and 1 to h . Introduce edges 110 through hh0 and 010, h0(h + 1), (h + 1)0. The resulting graph is a necklace (see Figure 3.1).

Figure 3.1:

Theorem 3.2.6. The only plane graphs with exactly two H-feasible embeddings are necklaces.

All other Halin graphs have exactly one H-feasible embedding. Thus, recognizing

Halin graphs among 3-connected planar graphs is easy. A simple characterization given in [36] is the following.

Theorem 3.2.7. Let G be a plane 3-connected graph of order n and size m. Then G is Halin if and only if one of its faces is of size m − n + 1. 20

A 1-factor of a graph G is a set of independent edges in G that span V (G). A graph is bicritical if for every pair of vertices u and v in G, G − {u, v} has a 1-factor.

Lov´aszand Plummer [23] showed that Halin graphs of even order has this property.

Theorem 3.2.8. Every even Halin graph H is minimal bicritical, i.e., H is bicritical and for each edge e of it, H − e is not bicritical.

Additionally, they gave a lower bound for the number of diﬀerent 1-factors contained in a Halin graph.

2 Theorem 3.2.9. If H is a Halin graph of even order n, then H has at least 3 (n − 1) number of 1-factors unless H is the graph of Figure 3.2. Moreover, this bound is sharp for all other Halin graphs.

Figure 3.2:

3.3 Hamiltonian properties

Bondy [5] proved that Halin graphs are Hamiltonian. In fact, Halin graphs possess strong Hamiltonian properties. Bondy (see [23]) further demonstrated the following result.

Theorem 3.3.1. All Halin graphs are 1-Hamiltonian. 21

For a Halin graph H, the minimum degree is δ(H) = 3. This together with the

Hamiltonicity of H guarantees that there are at least three Hamiltonian cycles in H.

Halin graphs enjoy further Hamiltonian properties. Skowro´nska [32] proved some of these for a larger class of graphs, called Halin-like graphs (and deﬁned in 3.4).

Theorem 3.3.2. Every Halin graph H = T ∪ C has the following properties:

(i) every edge of H belongs to a Hamiltonian cycle of H, i.e., H is 1-edge Hamilto- nian;

(ii) every pair of adjacent edges, one of which belongs to C, belongs to a Hamiltonian cycle of H.

A graph G is said to be uniformly Hamiltonian if each edge of G is contained in some Hamiltonian cycle and is avoided by another.

In addition to property (i) of Theorem 3.3.2 Skupien [35] proved

Theorem 3.3.3. Halin graphs are uniformly Hamiltonian.

Barefoot [1] showed another strong Hamiltonian property for Halin graphs.

Theorem 3.3.4. If H is a Halin graph, then H is Hamiltonian-connected.

Closely related with Hamiltonicity, is the notion of pancyclicity. Bondy and Lov´asz

[6] and, independently, Skowro´nska [33] announced their results regarding pancyclicity of Halin graphs.

Theorem 3.3.5. A Halin graph is almost pancyclic with the possible exception of an even cycle.

They also proved the following result which settles a conjecture of Malkevitch [25]. 22

Theorem 3.3.6. Let H = T ∪ C be a Halin graph, where T contains no vertex of degree 3. Then H is pancyclic.

The result of Theorem 3.3.5 is in a sense best possible, as showed by the following statement due to Malkevitch [25].

Theorem 3.3.7. If l is any even integer (l ≥ 4), then there exists a cubic Halin graph which is l-almost pancyclic.

3.4 Two variants of Halin graphs

It is easy to see that every interior face of a Halin graph H = T ∪ C has exactly one edge on the cycle C called exterior edge. Thus, as shown in [36], the intersection graph of the interior faces of H over its edge set (i.e., face intersection graph) is a 2-connected outer planar graph. Recall that a graph is outer planar if it can be embedded in the plane such that all of its vertices lie on the outer face. Con- versely, with every 2-connected outer planar graph G, we can associate a Halin graph whose face intersection graph is isomorphic to G. It is important to note that this correspondence between the class of Halin graphs and the class of 2-connected outer planar graphs through the operation of face intersection is not one-to-one, and there are 3-connected non-Halin graphs whose face intersection graph is 2-connected outer planar. Skowro´nska [32] called 3-connected plane graphs, that have 2-connected outer planar face intersection graphs, Halin-like graphs. Equivalently, such graph can be obtained by contracting exterior edges of a Halin graph. Hamiltonian properties of

Halin-like graphs proved by her are contained in the following theorem. 23

Theorem 3.4.1. Every Halin-like graph G is Hamiltonian and has, moreover the following properties:

(i) every edge of G belongs to a Hamiltonian cycle of G;

(ii) every two exterior edges of G which are adjacent to a vertex of degree 3 belong to a Hamiltonian cycle of G;

(iii) every pair of exterior and interior adjacent edges, which are in the same face belong to a Hamiltonian cycle of G;

(iv) G is 1-Hamiltonian.

Let u, v ∈ V (G). G is called locally u−v Hamiltonian-connected (or for uv ∈ E(G), locally doubly uv-edge Hamiltonian) if there are Hamiltonian u − v paths of G which cover two or more edges incident to both u and v. If this property holds for each pair of distinct vertices u and v (for each edge uv) of G, G is called locally doubly Hamiltonian- connected (locally doubly 1-edge Hamiltonian). These strong Hamiltonian properties exist in Halin-like graphs as proved by Skupien [35].

Theorem 3.4.2. Halin-like graphs are locally doubly Hamiltonian-connected.

Corollary 3.4.3. Halin-like graphs are locally doubly 1-edge Hamiltonian.

Corollary 3.4.4. Halin-like graphs are uniformly Hamiltonian.

Another variant of Halin graphs introduced by Skowro´nska and SysÃloin [34] was named skirted trees. A skirted tree S can be constructed from a plane tree T similarly as a Halin graph, except that T is allowed to have vertices of degree 2. Equivalently, a skirted tree can be obtained from a Halin graph by subdividing some of its interior edges. 24

A characterization of Hamiltonian skirted trees and those trees which have an embedding in the plane so that the resulting skirted trees are Hamiltonian are given in [34]. These characterizations are algorithmic and provide polynomial-time recog- nition algorithms for these graphs. Some of the results proved for Halin graphs can be extended to Hamiltonian skirted trees. For instance, the travelling salesman problem (TSP) has a polynomial time algorithm for Halin graphs (see [9]) and a similar approach can be used to solve the TSP on skirted trees. Chapter 4

3-Halin graphs

In chapter 3 we saw that Halin graphs possess a number of interesting Hamilto- nian properties. Now we generalize the notion of a Halin graph, call the generalized version a k-Halin graph, and investigate whether it inherits some of the Hamiltonian properties of a Halin graph. We shall see in this chapter, that the Hamiltonicity of k-Halin graphs steadily decreases as k increases. Even traceability is in general lost for k ≥ 3. 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.

4.1 k-Halin graphs

We generalize the notion of a 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. The cycle C is called outer cycle of G. The vertices and cycles in G − C are called inner vertices and, respectively, inner cycles of G.

25 26

A 0-Halin graph is a usual Halin graph. Moreover, the class of k-Halin graphs is contained in the class of (k + 1)-Halin graphs (k ≥ 0). Thus we get a nested sequence of generalized Halin graphs. We shall see in the following sections that, as expected, the Hamiltonicity of k-Halin graphs steadily decreases as k increases.

4.2 Hamiltonicity of 3-Halin graphs

The graph obtained from a Halin graph by deleting a vertex x of its outer cycle is called a reduced Halin graph [6]. The three neighbouring vertices of x, whose degrees reduce by one, are called the end-vertices of the reduced Halin graph. Lemma 1 of

[6] tells us the following.

Lemma 4.2.1. In any reduced Halin graph each pair of end-vertices is joined by a

Hamiltonian path.

Lemma 4.2.1 will allow us to contract any reduced Halin subgraph of a graph G to a single vertex of degree 3, whenever we study Hamiltonian properties of G.

A path in a k-Halin graph will be called an inner path, if it has its end-vertices on distinct inner cycles and no other vertex on any inner or outer cycle.

We call a k-Halin graph (k ≥ 1) simple if it is spanned by the union of all its inner paths and cycles and the outer cycle. Thus, a 1-Halin graph is simple if it has an inner cycle C1 (besides the outer cycle C), and is spanned by C ∪ C1.

Theorem 4.2.2. Every 1-Halin graph is Hamiltonian. 27

Proof. If the 1-Halin graph is also Halin, then it is Hamiltonian. Let now G be a 1-Halin graph with C1 and C as its inner and outer cycles respectively (see Figure 4.1).

Figure 4.1:

Let a be a vertex on C1 and let b∈ / C1 be a neighbor of a. If b∈ / C, the union of all paths from b to C, which do not contain a, is a tree Tb. This tree plus the edges on C between its leaves deﬁnes a reduced Halin graph Hb. We replace Hb by a single

0 vertex b ∈ C, adjacent with a ∈ C1. If b ∈ C, we keep the edge ab. After doing this with all vertices of C1, G reduces to a simple 1-Halin graph consisting of the two cycles C and C1, and of edges between the two cycles, such that the outer cycle has only vertices of degree 3 (see Figure 4.2). A Hamiltonian cycle in this graph is shown in Figure 4.2. This completes the proof.

Figure 4.2:

Remarks. A 1-Halin graph is not necessarily Hamiltonian connected. Indeed, Fig- ure 4.3 shows a bipartite 1-Halin graph G with 4 black and 4 white vertices. A path 28

between any pair of white vertices will have one more white vertex than black, so it cannot be Hamiltonian.

Figure 4.3: 1-Halin graph

A 2-Halin graph is not necessarily Hamiltonian. Indeed, Figure 4.4 shows a bipartite 2-Halin graph on 15 vertices. Such a graph has no Hamiltonian cycle.

Figure 4.4: 2-Halin graph

Recall that a graph admitting a spanning path is called traceable, and the path is called Hamiltonian.

Theorem 4.2.3. Every 2-Halin graph is traceable.

Proof. If the 2-Halin graph is also 1-Halin then, by Theorem 4.2.2, it is Hamil- tonian, hence traceable.

Let now G be a 2-Halin graph with inner cycles C1 and C2 and outer cycle C, as shown in Figure 4.5. 29

Figure 4.5:

Lemma 4.2.1 allows us to reduce G to a simple 2-Halin graph, that is the union of C, C1, C2, and the unique path P between C1 and C2 in G − C (possibly reduced to a vertex), plus edges between C and C1 ∪ C2 ∪ P (see Figure 4.6). Let a1 ∈ C1 and a2 ∈ C2 be the endpoints of P . We claim that there is a Hamiltonian path in

G between the neighbour b1 or c1 of a1 on C1 and the neighbour b2 or c2 of a2 on

C2. This is illustrated in Figure 4.6, where a path between b1 and b2 is realized. Accordingly, G is traceable.

Figure 4.6:

Remark. A 3-Halin graph is not necessarily traceable. Indeed, Figure 4.7 shows a

3-Halin bipartite graph G with 22 vertices coloured in two colours, 12 black and 10 white. 30

Figure 4.7: 3-Halin graph

4.3 Pancyclicity of 3-Halin graphs

As announced in the beginning of the Chapter, we show here that all 3-Halin graphs without inner vertices of degree 3 are pancyclic, thus extending the corresponding result of Bondy and Lov´asz[6] on Halin graphs. We shall make use of the following central result of [6].

Lemma 4.3.1. Every Halin graph is almost pancyclic. If the Halin graph H is l- almost pancyclic, then l is even and H must contain one of the three types of subgraphs depicted in Figure 4.8.

Figure 4.8: (l = 12)

Theorem 4.3.2. Every 3-Halin graph without inner vertices of degree 3 is pancyclic.

Proof. Let G be a 3-Halin graph without inner vertices of degree 3. There are at most 3 inner cycles in G. Choose an edge in each of them, such that no pair of edges 31

has a common point. Delete these edges. The resulting Halin graph H has at most

6 inner vertices of degree 3.

By Lemma 4.3.1, H is almost pancyclic. Assume cycles of length l are missing.

Then by Lemma 4.3.1, l is even and H must contain a reduced Halin graph of one of the types I, II, or III (Figure 4.8).

Suppose ﬁrst that l = 4. Then H must contain a reduced Halin graph H0 as described in Figure 4.9.

Figure 4.9:

The point x of H0 has degree 3. Hence it must belong in G to an edge e which has been deleted to obtain H. If the other endpoint of e is a vertex like x, i.e., a non-endpoint of a subgraph of H isomorphic to H0, then G has a cycle of length 4, and is therefore pancyclic. So, assume that the other endpoint of e is not a vertex like x. Since there are at most 3 edges like e, there are at most 3 vertices like x. But

4-almost pancyclic Halin graphs (see Figure 4.10) have more than 3 vertices like x if they are diﬀerent from the graph H00 of Figure 4.10. In case H = H00, the vertex o must on one hand have degree at least 4 in G, but can on the other hand be no endpoint of any further edge of G. Thus, in any case we obtain a contradiction.

Suppose now that l = 6. The smallest 6-almost pancyclic Halin graph is shown in Figure 4.11. This graph has 8 inner vertices of degree 3, so it cannot be H.

If l = 8, then, by Lemma 4.3.1, H must contain one of the reduced Halin subgraphs 32

Figure 4.10:

Figure 4.11: of Figure 4.12. Thus H has at least 6 inner vertices of degree 3, but they cannot be endpoints of only 3 edges in G, excepting the cases shown in Figure 4.13. In these cases, however, G has cycles of length 8, and is therefore pancyclic.

Figure 4.12:

If l ≥ 10, then the reduced Halin graph which must, by Lemma 4.3.1, appear in

H has at least 8 vertices of degree 3, which is impossible. This ends the proof.

The 37-Halin graph of Figure 4.14 has no cycle of length 4 and shows that not every k-Halin graph with no inner vertex of degree 3 must be pancyclic. So we are 33

Figure 4.13:

Figure 4.14: led to the following question.

Question Which is the maximal number k for which every k-Halin graph with no inner vertex of degree 3 is pancyclic? Chapter 5

15-Halin graphs

In chapter 4 we saw that hamiltonicity in k-Halin graphs steadily decreases, as k increases, and a 2-Halin graph is not necessarily Hamiltonian. In this chapter we shall see that the property of being Hamiltonian persists, however, for large values of k in cubic 3-connected k-Halin graphs. It will be shown that every cubic 3-connected 14-

Halin graph is Hamiltonian. The sharpness of our result is proved by presenting a non

Hamiltonian 15-Halin graph that is cubic and 3-connected. A variant of the famous example of Tutte [37] from 1946 which ﬁrst 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.

5.1 14-Halin graphs with connected core

In a k-Halin graph (k ≥ 1), the union of all inner cycles will be called core.A k-Halin graph (k ≥ 1) is called simple if it is spanned by the union of all its inner paths and cycles and the outer cycle.

Lederberg [21], Bos´ak[7] and Barnette found independently in 1966 the cubic

34 35

3-connected non-Hamiltonian graph with 38 vertices. This was not the ﬁrst graph of this kind, but rather small. At the end of a sequence of papers by Lederberg [22],

Butler [8], Barnette and Wegner [2], Okamura [27], [28], again Barnette [3], Holton and McKay [17], which appeared between 1966 and 1989, it was eventually proven that no smaller examples exist. This is stated in the following lemma.

Lemma 5.1.1. All cubic 3-connected planar graphs with at most 36 vertices are

Hamiltonian.

The next lemma is straightforward.

Lemma 5.1.2. Let G contain the fragment F shown in Figure 5.1. Replace F with the fragment F 0 and obtain a graph G0. If G0 is Hamiltonian then G is also Hamiltonian.

Figure 5.1:

Theorem 5.1.3. Every cubic 3-connected 14-Halin graph with connected core is

Hamiltonian.

Proof. Let G have k inner cycles (k ≤ 14) and denote by K the core of G. Then

K may only take one of the forms shown in Figure 5.2.

There are 7 possible values for k, namely 1, 3, 6, 7, 10, 12, 14. Our graph G is the union of K with the outer cycle C and with trees having all vertices of degree 3, 36

Figure 5.2: except for the leaves, one of which belongs to K and all the others to C. By Lemma

4.2.1, each such tree may be contracted to an edge between K and C without aﬀecting the Hamiltonicity of G. This transforms G into a simple 14-Halin graph (Figure 5.3).

Figure 5.3:

Without gaining Hamiltonicity, we may further reduce the order of G by observing that a repeated use of Lemma 5.1.2 decreases the number of consecutive edges between

K and C to at most two. Here, “consecutive edges”means that their vertices in K are adjacent. We obtain a graph as depicted in Figure 5.4, where in each case the maximal number, two, of consecutive edges is shown. In every case the order is at most 30. So, by Lemma 5.1.1, the graph is Hamiltonian. 37

Figure 5.4:

5.2 Almost k-Halin graphs

Let k ≥ 1. We say that a graph is almost k-Halin if it is obtained from a simple cubic 3-connected k-Halin graph, which is not (k −1)-Halin and has a connected core, by deleting a vertex of its outer cycle. The neighbours of that vertex are called the end-vertices of the almost k-Halin graph.

Lemma 5.2.1. In any almost 1-Halin graph each pair of end-vertices is joined by a

Hamiltonian path.

Proof. Let F be an almost 1-Halin graph with end-vertices b1, b2, b3. By deﬁni- tion, there exists a simple cubic 3-connected 1-Halin graph G and a vertex a on its outer cycle C, with neighbours b1, b2, b3, such that F = G − a (Figure 5.5). 38

Figure 5.5:

It is easy to see that G has a Hamiltonian cycle containing the path b1ab2 on its outer cycle. By deleting a, we get a Hamiltonian path between b1 and b2 in F .

Similarly we can ﬁnd Hamiltonian paths between b1, b3 and b2, b3 in F .

Lemma 5.2.2. In any almost 3-Halin graph, each pair of end-vertices is joined by a

Hamiltonian path.

Proof. The almost 3-Halin graph F deﬁnes as in the preceding proof a graph G and a vertex a ∈ G with the end-vertices b1, b2, b3 as neighbours. We use the notation of Figure 5.6. We apply Lemma 5.1.2 to reduce the number of consecutive edges between C and the core.

Case 1. We have only one edge between C1 and C.

Figure 5.6: 39

In this case C1 is a triangle and therefore may be contracted to a vertex of degree 3 without aﬀecting the Hamiltonicity and end-vertices of F . This reduces F to an almost 1-Halin graph, which has, by Lemma 5.2.1, a Hamiltonian path between each pair of end-vertices.

Case 2. There are exactly two edges between C1 and C.

Figure 5.7:

In this case C1 along with the pair of consecutive edges between it and C may be contracted, by Lemma 5.1.2, to a single edge, as shown in Figure 5.8.

Figure 5.8:

Thus, the original graph becomes a reduced Halin graph, which has, by Lemma

4.2.1, a Hamiltonian path between each pair of end-vertices. 40

Lemma 5.2.3. In any almost 6-Halin graph, each pair of end-vertices is joined by a

Hamiltonian path.

Proof. We use the same notation. Let Ci (i = 1, 2, 3) be cycles of the core, as shown in Figure 5.9.

After reducing the number of consecutive edges between C and the core, we have to consider two essentially diﬀerent cases for F , according to the position of b3 in the core.

Case 1. The end-vertex b3 belongs to C3. We have two subcases.

(i) There is one edge between C1 and C.

Figure 5.9:

In this case, the triangle C1 may be contracted to a single vertex, which transforms the graph to an almost 3-Halin graph. By Lemma 5.2.2, the latter has a Hamiltonian path between each pair of end-vertices.

(ii) There are two edges between C1 and C. As in Case 2 of Lemma 5.2.2, the graph may be contracted to the graph shown in

Figure 5.10 (middle). Keeping Hamiltonicity, we can successively contract a triangle to a vertex until the graph becomes an almost 1-Halin graph (see Figure 5.10). By

Lemma 5.2.1, this has a Hamiltonian path between each pair of end-vertices. 41

Figure 5.10:

Case 2. The end-vertex b3 belongs to C2. There are two possibilities.

(i) There is one edge between C1 and C.

Figure 5.11:

Then the graph may be contracted to an almost 3-Halin graph which, by Lemma

5.2.2, has a Hamiltonian path between each pair of end-vertices.

(ii) There are two edges between C and C1. The graph may be contracted to the graphs shown in succession in Figure 5.12.

A Hamiltonian path between any pair of end-vertices depends upon the number x of edges at place X in the graph. This directs us to consider the following cases.

(i) x = 1. 42

Figure 5.12:

Hamiltonian paths between end-vertices are shown in Figure 5.13.

Figure 5.13:

(ii) x = 2 (or 0).

Hamiltonian paths between end-vertices are shown in Figure 5.14.

Lemma 5.2.4. In any almost 10-Halin graph, each pair of end-vertices is joined by a Hamiltonian path.

Proof. We continue to use previous notation. Let Ci (i = 1, 2, 3, 4) be cycles of the core, as shown in Figure 5.15.

After reduction of consecutive edges we are led to consider two essentially diﬀerent cases for F .

Case 1. The end-vertex b3 belongs to C4. 43

Figure 5.14:

Figure 5.15:

We have two subcases to consider.

(i) There is one edge between C1 and C. In this case the graph may be contracted to an almost 6-Halin graph (Figure 5.16).

We apply Lemma 5.2.3.

Figure 5.16:

(ii) There are two edges between C1 and C. 44

Figure 5.17:

We contract the graph to an almost 3-Halin graph (see Figure 5.17) and apply

Lemma 5.2.2.

Case 2. The end-vertex b3 belongs to C3. We have two subcases to consider.

(i) There is one edge between C1 and C. The graph may be contracted to an almost 6-Halin graph (Figure 5.18).

Figure 5.18:

(ii) There are two edges between C and C1. Figure 5.19 shows how the graph may be contracted to an almost 3-Halin graph. 45

Figure 5.19:

Lemmas 5.2.1, 5.2.2, 5.2.3 and 5.2.4 allow us to contract any almost k-Halin graph

(k ∈ {1, 3, 6, 10}) to a vertex of degree 3, just as we can do with reduced Halin graphs.

Lemma 5.2.5. In any almost 7-Halin graph, at least two pairs of end-vertices are joined by Hamiltonian paths.

Proof. Following the notation of Lemma 5.2.1 and using Lemma 5.1.2, F will have at most 21 vertices, as shown in Figure 5.20.

Figure 5.20:

Assume F has only one pair of end-vertices, say b1, b2, joined by a Hamiltonian path. By Theorem 2.3.2 the Tutte triangle of Figure 5.21 has no Hamiltonian path from u to v. Then, by joining the end-vertices of F with the vertices of degree 2 of the

Tutte triangle appropriately (see Figure 5.22), we obtain a plane cubic 3-connected 46

non-Hamiltonian graph on at most 21 + 15 = 36 vertices. This contradicts Lemma

5.1.1. Hence F must have at least two pairs of end-vertices joined by Hamiltonian paths.

Figure 5.21:

Figure 5.22:

5.3 14-Halin graphs

Theorem 5.3.1. Let G be a cubic 3-connected k-Halin graph with core K, such that one component of K has at most 14 cycles and the number of cycles in any other component of K lies in {1, 3, 6, 10}. Then G is Hamiltonian. 47

Proof. First contract each tree with one leave a on an inner path or cycle and all the others on the outer cycle C to an edge from a to C. So, G becomes simple.

The graph having the components of K as vertices and the inner paths of G as edges between the corresponding vertices is a tree T .

Let x be a leave of T not corresponding to the component of K with at most

14 cycles mentioned in the statement. Thus, x corresponds to a component of K with 1, 3, 6 or 10 cycles. By Lemmas 4, 5, 6, 7, that component may be contracted to a vertex, and for the new graph the tree T has one vertex less. By continuing this procedure, we arrive at a simple 14-Halin graph with connected core, which is by

Theorem 5.1.3 Hamiltonian.

Theorem 5.3.2. Every cubic 3-connected 14-Halin graph is Hamiltonian.

Proof. Let G be a cubic 3-connected 14-Halin graph. We use the notation of

Theorem 5.3.1. If the components of K satisfy the requirements of Theorem 5.3.1, then G is Hamiltonian. If not, then K must have exactly two components, each with

7 cycles. We ﬁrst reduce G into a simple 14-Halin graph using Lemma 4.2.1. By

Lemma 5.2.5, an almost 7-Halin graph has at least two pairs of end-vertices joined by a Hamiltonian path. Let us call such pairs workable. G looks as shown in Figure

5.23 and is the union of two almost 7-Halin graphs H, H0 and a subgraph L depicted in Figure 5.24.

From any pair of end-vertices b1, b2 and b3 of H the subgraph L is traversed in a

0 0 0 0 unique way, leading to a unique pair of end-vertices b1, b2, b3 of H . This obviously establishes a one-to-one correspondence between the pairs (b1, b2), (b2, b3), (b3, b1) and

0 0 0 0 0 0 the pairs (b1, b2), (b2, b3), (b3, b1) (see Figure 5.25.) Since there are at least two workable pairs of end-vertices on each side, at least 48

Figure 5.23:

Figure 5.24: one workable pair on one side is in correspondence with a workable pair on the other side. This guarantees the existence of a Hamiltonian cycle in G.

5.4 15-Halin graphs

Theorem 5.4.1. There are 3-connected 15-Halin graphs which are not Hamiltonian.

Proof. Figure 5.26 depicts a cubic 3-connected 15-Halin graph G on 42 vertices that comprises two Tutte triangles T , T 0, each containing a component of the core with 7 cycles, and one additional component with one cycle in the middle. Since there is no Hamiltonian path between a and b in T , the workable pairs in T are a, c and b, c only. And the workable pairs in T 0 are a0, c0 and b0, c0. 49

Figure 5.25:

Figure 5.26 also shows that any pair of disjoint paths from a workable pair of T goes through the middle inner cycle in a unique way and eventually ends at the pair a0, b0 of T 0. Hence G is not Hamiltonian. 50

Figure 5.26:

Final remarks. Theorems 5.3.2 and 5.4.1 settle the problem of ﬁnding the maximal k for which all cubic 3-connected k-Halin graphs are Hamiltonian. A variant of the famous example of Tutte [37] from 1946 which ﬁrst demonstrated that cubic

3-connected planar graphs may not be Hamiltonian, is a 21-Halin graph. The cubic

3-connected planar non-Hamiltonian graph of Lederberg [21], Bos´ak[7] and Barnette, which has smallest order, is 53-Halin.

It would be interesting to investigate the analogous question about traceability.

However, the exact calculation of the maximal number k up to which all cubic 3- connected k-Halin graphs are traceable seems out of reach, as the corresponding, more general problem of ﬁnding the maximal order up to which all cubic 3-connected graphs are traceable is very far from being solved. Chapter 6

Conclusion

Halin graphs have received considerable attention from the research community in the past. In this thesis we have investigated how far we can extend the notion of a

Halin graph such that hamiltonicity is preserved. A k-Halin graph G can be obtained from a Halin graph H = T ∪ C by adding edges while keeping planarity, joining non leaves of T such that G − C has at most k cycles.

We proved that a 1-Halin graph is still Hamiltonian, but not Hamiltonian connected, 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, the pancyclicity of Halin graphs with no inner vertex of degree 3 remains true even for 3-Halin graphs. We presented a 37-Halin graph with no inner vertex of degree 3 that contains no cycle of length 4. This invites further research to compute the maximal number k for which every k-Halin graph without inner vertex of degree

3 is pancyclic.

51 52

The property of being Hamiltonian persisted for large values of k in cubic 3- connected k-Halin graphs. We proved that every cubic 3-connected 14-Halin graph is Hamiltonian. The sharpness of our result is guaranteed by presenting a non-

Hamiltonian cubic 3-connected 15-Halin graph. It would be interesting to investigate the analogous question about traceability in the same class of graphs. However, the exact calculation of the maximal number k up to which all cubic 3-connected k-Halin graphs are traceable seems out of reach, as the corresponding, more general problem of ﬁnding the maximal order up to which all cubic 3-connected graphs are traceable is very far from being solved. Bibliography

[1] C. A. Barefoot, Hamiltonian connectivity of the Halin graphs, Congressus numerantium 58 (1987), 93-102.

[2] D. Barnette and G. Wegner, Hamiltonian circuits in simple 3-polytope with up to 26 vertices, Israel J. Math. 19 (1974), 212-216.

[3] D. Barnette, Every simple 3-polytope with 34 vertices is Hamiltonian, Discrete Math. 162 (1986), 1-20.

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