Some cyclic properties of graphs with local Ore-type conditions

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Some cyclic properties of graphs

with local Ore-type conditions

Department of Mathematics, Linköping University Jonas Granholm

LiTH-MAT-EX--2016/04--SE

Credits: 30 hp Level: A

Supervisor: Armen Asratian,

Department of Mathematics, Linköping University Examiner: Carl Johan Casselgren,

Department of Mathematics, Linköping University Linköping: August 2016

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Abstract

A Hamilton cycle in a graph is a cycle that passes through every vertex of the graph. A graph is called Hamiltonian if it contains such a cycle.

In this thesis we investigate two classes of graphs, defined by local criteria. Graphs in these classes, with a simple set of exceptions K, were proven to be Hamiltonian by Asratian, Broersma, van den Heuvel, and Veldman in 1996 and by Asratian in 2006, respectively.

We prove here that in addition to being Hamiltonian, graphs in these classes have stronger cyclic properties. In particular, we prove that if a graph G belongs to one of these classes, then for each vertex x in G there is a sequence of cycles such that each cycle contains the vertex x, and

• the shortest cycle in the sequence has length at most 5;

• the longest cycle in the sequence is a Hamilton cycle (unless G belongs to the set of exceptions K, in which case the longest cycle in the sequence contains all but one vertex of G);

• each cycle in the sequence except the first contains all vertices of the previous cycle, and at most two other vertices.

Furthermore, for each edge e in G that does not lie on a triangle, there is a sequence of cycles with the same three properties, such that each cycle in the sequence contains the edge e.

Keywords:

Hamiltonian, pancyclic, vertex pancyclic, edge pancyclic, cycle extendable, local conditions, Ore-type conditions

URL for electronic version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-129213

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Sammanfattning

En Hamiltoncykel i en graf är en cykel som går genom varje hörn i grafen. En graf kallas Hamiltonsk om den innehåller en sådan cykel.

I detta arbete undersöks två klasser av grafer som definieras av lokala villkor. Grafer i dessa klasser, förutom en enkel mängd undantagna grafer K, har bevisats vara Hamiltonska av Asratian, Broersma, van den Heuvel och Veldman 1996 respektive av Asratian 2006.

Vi bevisar här att grafer i dessa klasser inte bara är Hamiltonska, utan har starkare cykliska egenskaper. Mer specifikt bevisar vi att om en graf G tillhör en av dessa klasser, så gäller för varje hörn x i G att det finns en följd av cykler sådan att varje cykel innehåller hörnet x, och

• den kortaste cykeln i följden har som mest längden 5

• den längsta cykeln i följden är en Hamiltoncykel (om inte G tillhör undan-tagsmängden K, i vilket fall den längsta cykeln i följden innehåller alla hörn i G utom ett)

• varje cykel i följden utom den första innehåller alla hörn i den föregående cykeln, och som mest två hörn till.

Dessutom gäller för varje kant e i G som inte ligger på en triangel att det finns en följd av cykler med samma tre egenskaper, sådan att varje cykel i följden innehåller kanten e.

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Acknowledgements

I would like to thank my supervisor Armen Asratian for introducing me to this interesting subject and for inspiring me to explore it. Without his help and encouragement this thesis would never be finished, and without his many corrections and valuable suggestions it would never be understood by anyone. I would also like to thank my examiner Carl Johan Casselgren and my opponent Emil Karlsson for their proofreading and suggestions. Finally, I would like to thank my family and especially my wife for their unending support.

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Figures

2.1 A graph, and the 1- and 2-ball around a vertex v . . . . 7 3.1 The graph in Hamilton’s game . . . 11 3.2 A solution to Hamilton’s game . . . 12 3.3 A nonhamiltonian graph in which every ball that is not the whole

graph is Hamiltonian . . . 13 4.1 A nonpancyclic and a nonhamiltonian graph . . . 21 4.2 Two graphs with vertices and edges that do not lie on any cycle of

length less than 5 . . . 23 4.3 The graph K₂∨ K₃∨ K₃∨ K₁ . . . 23 4.4 A graph with a cycle that cannot be expanded by less than three

vertices . . . 24 4.5 A graph in which every edge lies on a triangle . . . 35

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Chapter 1

Introduction

A Hamilton path in a graph is a path that passes through every vertex of the graph, and a Hamilton cycle in a graph is a cycle that passes through every vertex of the graph. A graph containing a Hamilton cycle is called Hamiltonian. Hamiltonicity is one of the most fundamental notions in graph theory and has been studied extensively (see e.g. [20, 21]). Problems related to Hamiltonicity arise in different areas of mathematics.

Example [14]. Given nonnegative integral vectors R = (r₁, . . . , r_m) and S = (s₁, . . . , s_n), let A(R, S) denote the set of all m × n-matrices of 0’s and 1’s that have ri 1’s in the ith row and sj 1’s in the jth column, for each i = 1, . . . , m

and j = 1, . . . , n. An interchange of a matrix A ∈ A(R, S) is a transformation that replaces a 2 × 2 submatrix (1 0

0 1) with a 2 × 2 submatrix (0 11 0) or vice versa.

Suppose that A(R, S) 6= ∅.1 Is it possible to generate the matrices from A(R, S) without repetitions in such a way that, given the first, each subsequent matrix is obtained from its predecessor by a single interchange? In terms of graphs this problem can be reformulated as follows: Does there exist a Hamilton path or cycle in the graph G(R, S) whose vertices are the matrices in A(R, S), and for A, B ∈ A(R, S) there is an edge joining A and B if and only if B can be obtained from A by a single interchange?

In general, certain algebraic or combinatorial objects can be generated without repetition using only one special operation if and only if a related graph contains a Hamilton cycle or path.

1_{A necessary and sufficient condition for A(R, S) 6= ∅ was found in [18, 29].}

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2 Chapter 1. Introduction

The problem of determining if a graph is Hamiltonian is NP-complete [25], which means that no efficient method of finding Hamilton cycles is likely to exist. There are, however, many conditions that have been proven to imply Hamiltonicity. A classical result giving such a condition is Dirac’s Theorem [17], which states that a graph G with at least three vertices is Hamiltonian if each vertex v ∈ V (G) has degree d(v) ≥ |V (G)|/2. This result was later generalized by Ore [28] to give the following: A graph G with at least three vertices is Hamiltonian if every pair of nonadjacent vertices u, v ∈ V (G) has degree sum

d(u) + d(v) ≥ |V (G)|.

All Hamiltonian graphs – and thus all graphs satisfying the condition of Ore’s Theorem – are 2-connected, that is, they cannot be split into two disconnected parts by removing a single vertex. Jung [24] showed that the limit |V (G)| in Ore’s Theorem can be relaxed to |V (G)| − 1, as long as G remains 2-connected and does not belong to the class K consisting of all complete bipartite graphs

K_p,p+1 and all graphs obtained from K_p,p+1 by joining some vertices in the smaller part of the bipartition. A few years earlier, the same result had been proven by Nash-Williams [27] for regular graphs, though in that case the set K of exceptions is not needed.

A disadvantage with the above results is that they only apply to graphs with diameter 1 or 2, that is, graphs where the distance between any two vertices does not exceed 2. Furthermore, every graph satisfying one of these conditions is dense, that is, the number of edges in the graph is proportional to the square of the number of vertices.

One method, suggested by Asratian and Khachatrian (see e.g. [5, 6, 7]), that has received attention lately is to replace global conditions with local analogues. This means that instead of checking the whole graph at once, it suffices to consider a large number of balls with small radius, where a ball of radius r around a vertex v is the subgraph induced by the set of vertices at distance at most r from v. An advantage of such an approach is that multiple computers can process different parts of the graph simultaneously. Another advantage of these localized theorems is that they, unlike their global equivalents, apply to families of sparse graphs with large diameter.

Asratian and his colleagues have shown [5, 6, 4, 3] that the results of Dirac, Ore, Jung, and Nash-Williams can be localized in such a fashion in different ways. For example, the following, from [6], is a localization of Dirac’s Theorem: A connected graph G with at least three vertices is Hamiltonian if each vertex

v ∈ V (G) has degree d(v) ≥ |M₃(v)|/2, where M₃(v) is the set of vertices in G that are at distance at most three from v.

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Chapter 1. Introduction 3

Many sufficient conditions for Hamiltonicity also imply stronger properties. In 1971 Bondy [10] proved that every graph G that satisfies the conditions of Ore’s Theorem, except the complete bipartite graphs Kn,n, are pancyclic, that

is, they contain a cycle of every length from 3 up to |V (G)|. This prompted him to make a famous metaconjecture [11]: “Almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. (There may be a simple family of exceptional graphs.)” Since then, many conditions that imply Hamiltonicity have been proven to imply pancyclicity.

Closely related to pancyclicity is the concept of cycle extendability. A graph with at least one cycle is cycle extendable if for every nonhamiltonian cycle C in the graph there is a cycle C0 containing all vertices of C and exactly one other vertex. Some classes of cycle extendable graphs were found by Hendry [22] and Asratian [2]. Not every graph that satisfies the condition of Ore’s Theorem is cycle extendable. However, Bondy [12] noted that for every nonhamiltonian cycle C in a graph satisfying the conditions of Ore’s Theorem there is a cycle C0 containing all vertices of C and one or two other vertices.

In this thesis we prove a similar property for two classes of graphs, defined by local criteria. Graphs in these classes, with the set of exceptions K described above, were proven to be Hamiltonian by Asratian, Broersma, van den Heuvel, and Veldman [4] in 1996 and by Asratian [3] in 2006, respectively. We show that not all graphs in these classes are pancyclic or cycle extendable, but every graph in these classes possess some related cyclic properties. In particular, we prove that if a graph G belongs to one of these classes, then for each vertex x in G there is a sequence of cycles such that each cycle contains the vertex x, and

• the shortest cycle in the sequence has length at most 5;

• the longest cycle in the sequence is a Hamilton cycle (unless G belongs to the set of exceptions K, in which case the longest cycle in the sequence contains all but one vertex of G);

• each cycle in the sequence except the first contains all vertices of the previous cycle, and at most two other vertices.

Furthermore, for each edge e in G that does not lie on a triangle, there is a sequence of cycles with the same three properties, such that each cycle in the sequence contains the edge e.

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Chapter 2

Fundamentals

This chapter will define notions used in the thesis. For a more thorough intro-duction to graph theory, see [13]. For more information on NP-completeness, see [19].

2.1 Basic definitions

Definition 2.1.1. A graph G is a pair V (G), E(G) consisting of a set V (G) of vertices and a set E(G) of edges, where each edge connects two distinct vertices and no two vertices are connected by more than one edge. Two vertices are adjacent if they are connected by an edge.

Definition 2.1.2. A complete graph is a graph where every pair of vertices is connected with an edge. The complete graph with n vertices is denoted Kn.

Definition 2.1.3. Two vertices are independent if they are not connected by an edge. An independent set S of vertices is a subset S ⊆ V (G) such that all pairs of vertices in S are independent.

Definition 2.1.4. The complement of a graph G, denoted G, is a graph such that V G = V (G), and two vertices in G are joined by an edge if and only if those vertices are not joined by an edge in G.

Thus the complement of a complete graph K_n is an empty graph K_n without any edges. For any graph G, if we let V0= V (G) = V G and E0_{= E(G)∪E G,}

the graph G0= (V0, E0) is complete.

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6 Chapter 2. Fundamentals

Definition 2.1.5. The degree of a vertex v ∈ V (G), denoted d(v), is the number of edges that are incident with v.

Definition 2.1.6. A k-regular graph is a graph where every vertex has degree k. A regular graph is a graph that is k-regular for some k.

Definition 2.1.7. If X and Y are two disjoint subsets of V (G), then the number of edges connecting one vertex in X and one vertex in Y is denoted e(X, Y ). Definition 2.1.8. A graph G0 is a subgraph of the graph G if V (G0) ⊆ V (G) and E(G0) ⊆ E(G). If S ⊆ V (G), then the graph G0where V (G0) = S and E(G0) contains every edge in E(G) that joins two vertices of S is called the subgraph of

G induced by S. The subgraph induced by the set V (G) \ S is denoted G − S.

Definition 2.1.9. A path is a nonempty graph or subgraph of the form P = (V, E) with V = {v₀, v₁, . . . , vn} and E = {v0v1, v1v2, . . . , vn−1vn} where all vi

are distinct. We say that P joins the vertices v₀ and v_n, or that P is a v0−vn

-path. The number of edges in a path is called the length of the -path. A path containing n vertices is called an n-path, and is denoted Pn.

Remark 2.1.1. Note that an n-path Pn has length n − 1.

Definition 2.1.10. A cycle is a nonempty graph or subgraph of the form

C = (V, E) with V = {v₀, v₁, . . . , v_n} and E = {v₀v₁, v₁v₂, . . . , v_n−1v_n, v_nv₀} where all v_i are distinct. The number of edges in a cycle is called the length of the cycle. A cycle containing n vertices is called an n-cycle, and is denoted Cn.

A cycle of length three is called a triangle, and a cycle of length four is called a square.

We will usually represent a path as a sequence v₀v₁v₂· · · v_n of its vertices. Similarly we represent a cycle as a sequence v0v1v2· · · vnv0of its vertices. To

indi-cate that a path is traversed in a particular direction we use the notation ~P , and

we denote the same path in the reverse direction by P

~

. When we have specified a direction of a path, we use the notation v+

i to denote the successor of vi on ~P

and v_i−to denote the predecessor. If ~P is the path v₀v₁v₂· · · v_ndirected from v₀ to v_n, then we denote a subpath v_iv_i+1· · · v_j of ~P from v_ito v_j by v_iP v~ _j, and in the other direction it is denoted by v_jP

~

v_i. Analogous notation is used for cycles. Definition 2.1.11. Two vertices are connected if there is a path joining them. A graph is connected if all pairs of vertices are connected, otherwise it is disconnected.

If S is a set of vertices such that removing the vertices of S and all edges incident with those vertices from G makes G disconnected, S is called a vertex

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2.1. Basic definitions 7

v G₁(v): v G₂(v): v

Figure 2.1: A graph, and the 1- and 2-ball around a vertex v.

cut set. If S = {v}, then v is called a cut vertex. A graph is k-connected if it contains at least k + 1 vertices but no vertex cut set with less than k vertices. A maximal connected subgraph of G is called a component of G. The number of components in G is denoted ω(G).

Definition 2.1.12. A graph G is 1-tough if ω(G − S) ≤ |S| for any proper subset S ⊂ V (G), that is, if removing k vertices cannot split G into more than

k disconnected parts.

Definition 2.1.13. The distance between two vertices u and v, denoted

d(u, v), is the length of the shortest path joining u and v. If u and v are

not connected by any path, then d(u, v) is infinite.

Definition 2.1.14. The diameter of a graph is the longest distance between any two vertices in the graph. If a graph is disconnected, then its diameter is infinite.

Definition 2.1.15. The neighbourhood of a vertex v, denoted N (v), is the set of vertices adjacent to v. The set of vertices at distance at most r from a vertex v (including v itself) is denoted Mr(v). The ball of radius r around v,

or just the r-ball around v, denoted Gr(v), is the subgraph induced by Mr(v).

In Figure 2.1 we see a graph and two balls of differing radii in that graph. Definition 2.1.16. A graph G is bipartite if the vertices of G can be parti-tioned into two sets such that every edge of G has one end in each of the sets.

A bipartite graph in which every two vertices from different parts of the bipar-tition are adjacent is called complete bipartite. The complete bipartite graph with m vertices in one of its parts and n vertices in the other is denoted Km,n.

Remark 2.1.2. A graph is bipartite if and only if it does not contain any cycle of

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8 Chapter 2. Fundamentals

Definition 2.1.17. A matching in a graph G is a subset M ⊆ E(G) such that no two edges in M have a common endpoint. A matching is perfect if every vertex of G is the endpoint of an edge in the matching.

Definition 2.1.18. The union of two graphs G₁ and G₂, denoted G1∪ G2,

is the graph with vertex set V (G₁) ∪ V (G₂) and edge set E(G₁) ∪ E(G₂). The intersection of two graphs G1 and G2, denoted G1∩ G2, is the graph with

vertex set V (G₁) ∩ V (G₂) and edge set E(G₁) ∩ E(G₂). If G₁∩ G₂= ∅, then G₁ and G2 are said to be disjoint.

Definition 2.1.19. The join of two disjoint graphs G1and G2, denoted G1∨G2,

is the graph obtained from G₁ ∪ G₂ by joining each vertex of G₁ to each vertex of G₂by an edge. The sequential join of a sequence of disjoint graphs

G₁, G₂, . . . , G_n, denoted G1 ∨ G2 ∨ · · · ∨ Gn, is the graph obtained from

G₁∪ G₂∪ · · · ∪ G_n by adding edges joining each vertex of G₁to each vertex of

G₂, each vertex of G₂ to each vertex of G₃, and so on.

2.2 NP-completeness

Many problems in graph theory are decision problems; that is, questions that can be answered by yes or no, such as

• Can G be coloured with four different colours such that no adjacent vertices have the same colour?

• Does G contain a cycle?

• Does G contain a cycle that passes through every vertex?

Some of these questions can be answered in polynomial time, which means that the time it takes to answer the question1 can be described by a polynomial of the size of the input. The time it takes to check if a graph contains a cycle, for example, is a linear function of the size of the graph, so if the size is doubled, then the time is (approximately) doubled as well. For other problems, the time it takes might triple or quadruple when the size of the input doubles. These problem are said to be in the complexity class P (which simply stands for polynomial time).

1_{Technically measured in number of operations a computer must do, but approximately}

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2.2. NP-completeness 9

For many problems, no polynomial algorithm is known. This includes finding a cycle passing through every vertex, and deciding if there is a four-colouring where no adjacent vertices have the same colour. The best known algorithms for solving these problems take exponential time, which means that the time it takes to solve the problem is doubled whenever the size of the input is increased by a certain fixed number. For large inputs, these problems quickly become infeasible to solve using general methods, sometimes literally taking millennia to solve on typical computers. When a solution is found however (e.g., a cycle or a colouring), it can be verified in polynomial time. This complexity class is called NP (it stands for nondeterministic polynomial time, which comes from the formal definition of NP). It is currently unknown whether or not every problem whose solution can be verified in polynomial time can also be solved in polynomial time, i.e., whether P = NP or not.

A problem in NP is called NP-complete if any other problem in NP can, in polynomial time, be transformed in a way such that it becomes an instance of the first problem. Informally speaking, a problem is NP-complete if it is at least as hard as any other problem in NP. Thus, if there would be a polynomial algorithm for some NP-complete problem, then every problem for which a solution can be verified in polynomial time would also be solvable in polynomial time. These concepts have been studied extensively, and there are many problems known to be NP-complete. Since no one has found a polynomial algorithm that solves any NP-complete problem, despite much research, it is likely that P 6= NP, which would mean that every NP-complete problem has instances that cannot be solved efficiently.

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Chapter 3

Hamiltonicity and related

concepts

3.1 Hamiltonian graphs

In 1857 Sir William Rowan Hamilton introduced the following game: He drew a graph representing the edges of a dodecahedron (see Figure 3.1) and labeled each vertex with the name of a European city, and had the edges of the graph represent roads between the cities. Then he asked if it was possible to start in one city, visit each city exactly once, and finally return to the first city. A solution is not hard to find (see Figure 3.2), but the more general form of the puzzle – finding a cycle that passes through all vertices of an arbitrary graph – has proven to be trickier.

Definition 3.1.1. A cycle that passes through every vertex of a graph is called a Hamilton cycle. A graph containing a Hamilton cycle is Hamiltonian. A path that passes through every vertex of a graph is called a Hamilton path.

Figure 3.1: The graph in Hamilton’s game.

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12 Chapter 3. Hamiltonicity and related concepts

Figure 3.2: A solution to Hamilton’s game.

Hamilton cycles are important in many applications, and have been studied extensively (see e.g. [20, 21]). However, in 1972 Karp proved the following: Theorem 3.1.1 (Karp [25]). The problem of determining if a graph is Hamil-tonian is NP-complete.

Since the problem of finding a Hamilton cycle in a graph is NP-complete, a lot of the research has been focused on finding ways to determine if a graph contains a Hamilton cycle without actually doing the hard work of finding it. It is easy to see that every Hamiltonian graph must be 2-connected, so a graph that is not 2-connected cannot contain a Hamilton cycle. This condition was strengthened by Chvátal:

Theorem 3.1.2 (Chvátal [16]). If G is Hamiltonian, then G is 1-tough. A classical results that guarantees Hamiltonicity is the following:

Theorem 3.1.3(Dirac [17]). Let G be a graph with at least three vertices such that d(v) ≥ |V (G)|/2 for every vertex v ∈ V (G). Then G is Hamiltonian. This was later generalized by Ore:

Theorem 3.1.4 (Ore [28]). Let G be a graph with at least three vertices such that d(u) + d(v) ≥ |V (G)| for every pair of nonadjacent vertices u, v ∈ V (G). Then G is Hamiltonian.

The condition in Ore’s Theorem can be relaxed, if we allow a set of exceptions: Theorem 3.1.5 (Jung [24], Nara [26]). Let G be a 2-connected graph such that d(u) + d(v) ≥ |V (G)| − 1 for every pair of nonadjacent vertices u, v ∈ V (G). Then G is Hamiltonian unless G ∈ K, where

K = { G : K_p,p+1⊆ G ⊆ K_p∨ K_p+1 for some p ≥ 2 }.

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3.1. Hamiltonian graphs 13

Figure 3.3: A nonhamiltonian graph in which every ball that is not the whole graph is Hamiltonian.

Theorem 3.1.6 (Nash-Williams [27]). Let G be a 2-connected regular graph such that d(v) ≥ (|V (G)|−1)/2 for each vertex v ∈ V (G). Then G is Hamiltonian. Asratian and his colleagues have obtained local analogues of these four theorems [5, 6, 4, 3]. The main idea of their method is to use the structure of balls of small radii. It is not as simple as checking if the balls are Hamiltonian, however; every ball in the graph in Figure 3.3 that is not the whole graph is Hamiltonian, but the whole graph is not. The following is a localization (and generalization) of Ore’s Theorem:

Theorem 3.1.7 (Asratian–Khachatrian [6]). Let G be a connected graph with at least three vertices such that for every triple u, w, v with d(u, v) = 2 and

w ∈ N (u) ∩ N (v) the following property holds:

d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)|.

Then G is Hamiltonian.

Theorem 3.1.7 was later generalized to the following, using the set of excep-tions K from Jung’s Theorem:

Theorem 3.1.8 (Asratian–Broersma–van den Heuvel–Veldman [4]). Let G be a connected graph with at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following two properties hold:

d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)| − 1

and |N (u) ∩ N (v)| ≥ 2. Then G is Hamiltonian unless G ∈ K.

Theorem 3.1.8 is not a generalization of Jung’s Theorem, as the condition |N (u) ∩ N (v)| ≥ 2 is too restrictive; for example, the graph C₅ is excluded. A localization that generalizes Jung’s Theorem was obtained by Asratian:

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Theorem 3.1.9 (Asratian [3]). Let G be a connected graph with at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following property holds:

d(u) + d(v) ≥ |M₂(w)| − 1,

and furthermore every 2-ball in G is 2-connected. Then G is Hamiltonian unless

G ∈ K.

We will extend the last two results in Chapter 4.

3.2 Related concepts

Many of the sufficient conditions for Hamiltonicity, such as Ore’s Theorem, do not just imply Hamiltonicity, but also stronger properties like pancyclicity. Definition 3.2.1. A graph G is pancyclic if it contains a cycle of every length from 3 up to |V (G)|.

In 1971 Bondy proved the following:

Theorem 3.2.1 (Bondy [10]). Let G be a graph satisfying the conditions of Ore’s Theorem. Then G is pancyclic unless G is a complete bipartite graph Kn,n

for some n ≥ 2.

This prompted him to make a famous metaconjecture [11]: “Almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. (There may be a simple family of exceptional graphs.)” Since then, many conditions that imply Hamiltonicity have been proven to imply pancyclicity; for example, the one in Theorem 3.1.7:

Theorem 3.2.2 (Asratian–Sarkisian [8]). Let G be a connected graph with at least three vertices such that for every triple u, w, v with d(u, v) = 2 and

d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)|.

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3.2. Related concepts 15

Two notions that are even stronger than pancyclicity are vertex and edge pancyclicity:

Definition 3.2.2. A graph G is vertex pancyclic if for every vertex v ∈ V (G) there is a cycle of every length from 3 up to |V (G)| containing v.

Definition 3.2.3. A graph G is edge pancyclic if for every edge e ∈ E(G) there is a cycle of every length from 3 up to |V (G)| containing e.

Xiao Tao Cai proved the following theorem:

Theorem 3.2.3 (Xiao Tao Cai [15]). Let G be a graph with at least four vertices satisfying the conditions of Ore’s Theorem. Then each vertex of G lies on a cycle of every length from 4 up to |V (G)| unless G is a complete bipartite graph Kn,n

for some n ≥ 2. Thus G is vertex pancyclic if and only if every vertex of G lies on a triangle.

This result was generalized to the set of graphs satisfying the conditions of Theorem 3.1.7:

Theorem 3.2.4 (Asratian–Sarkisian [9]). Let G be a connected graph with at least four vertices such that for every triple u, w, v with d(u, v) = 2 and

d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)|.

Then each vertex of G lies on a cycle of every length from 4 up to |V (G)| unless

G is a complete bipartite graph K_n,n for some n ≥ 2. Thus G is vertex pancyclic if and only if every vertex of G lies on a triangle.

It follows from Remark 2.1.2 that no bipartite graph can be pancyclic. It is nevertheless interesting to study cyclic properties of bipartite graphs, and the following definition is a bipartite analogue of pancyclicity:

Definition 3.2.4. A bipartite graph G is bipancyclic if it contains a cycle of every even length from 4 up to |V (G)|.

The concepts of vertex bipancyclicity and edge bipancyclicity can naturally be defined analogously to Definitions 3.2.2 and 3.2.3.

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Closely related to pancyclicity is the concept of extending cycles.

Definition 3.2.5. A graph G is cycle extendable if G contains at least one cycle, and for every nonhamiltonian cycle C_n of length n in G there is a cycle

Cn+1of length n + 1 containing every vertex of Cn.

Not every graph that satisfies the condition of Ore’s Theorem is cycle extend-able. However, Bondy noted the following property, which we will generalize in Chapter 4:

Theorem 3.2.5 (Bondy [12]). Let G be a graph satisfying the conditions of Ore’s Theorem. Then for every nonhamiltonian cycle C_n of length n in G there is a cycle C_n+`of length n + `, where 1 ≤ ` ≤ 2, such that V (C_n) ⊂ V (C_n+`).

Hendry [22] found some classes of cycle extendable graphs. In particular he found a criterion for graphs satisfying the conditions of Ore’s Theorem to be cycle extendable. Another class of cycle extendable graphs was found by Asratian:

Theorem 3.2.6 (Asratian [2]). Let G be a connected graph such that for every vertex v ∈ V (G), the ball G₁(v) satisfies the conditions of Ore’s Theorem. Then

G is cycle extendable.

As for pancyclicity, no bipartite graphs can be cycle extendable. However, in [23] Hendry introduced the notion of bi-cycle extendability.

Definition 3.2.6. A bipartite graph G is bi-cycle extendable if G contains at least one cycle, and for every nonhamiltonian cycle C_n of length n in G there is a cycle Cn+2of length n + 2 containing every vertex of Cn.

Hamiltonian graph theory is not only focused on cycles however. A related property that has also received attention is based on Hamilton paths.

Definition 3.2.7. A graph G is Hamilton-connected if for every pair of vertices x, y ∈ V (G) there is a Hamiltonian x–y-path.

It is easy to see that every Hamilton-connected graph with at least three vertices is Hamiltonian. Furthermore, every Hamilton-connected graph with at least four vertices must be 3-connected, since any two vertices are the endpoints of a Hamilton path. If we add this condition to Theorem 3.1.7 we get the following:

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3.2. Related concepts 17

Theorem 3.2.7 (Asratian [1]). Let G be a 3-connected graph such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following property holds:

d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)|.

Then G is Hamilton-connected unless G ∈ M, where

M = { G : K_p,p⊆ G ⊆ K_p∨ K_p for some p ≥ 3 }.

The relaxed conditions of Theorem 3.1.8 do not imply Hamilton-connectedness, as demonstrated by the graphs obtained by deleting a perfect matching from

K_p,pand K_p∨ K_pfor p ≥ 4. They do however imply a weaker property, which we will strengthen in Chapter 4:

Theorem 3.2.8 (Asratian–Broersma–van den Heuvel–Veldman [4]). Let G be a connected graph with at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following two properties hold:

d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)| − 1

and |N (u) ∩ N (v)| ≥ 2. Then every pair of vertices x, y with d(x, y) ≥ 3 is connected by a Hamilton path of G.

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Chapter 4

Results

As mentioned earlier, in [4] and [3] generalizations of the theorems of Dirac [17], Ore [28], Jung [24], and Asratian–Khachatrian [6] were found (see Theorems 3.1.8 and 3.1.9). In this chapter we strengthen these results. For the reader’s con-venience we repeat the theorems here.

Theorem A (Asratian–Broersma–van den Heuvel–Veldman [4]). Let G be a connected graph with at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following two properties hold:

(A.1) d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)| − 1 (A.2) |N (u) ∩ N (v)| ≥ 2.

Then G is Hamiltonian unless G ∈ K, where

K = { G : Kp,p+1⊆ G ⊆ Kp∨ Kp+1 for some p ≥ 2 }.

Theorem B (Asratian [3]). Let G be a connected graph with at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following property holds:

(B.1) d(u) + d(v) ≥ |M₂(w)| − 1, and furthermore

(B.2) every 2-ball in G is 2-connected. Then G is Hamiltonian unless G ∈ K.

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20 Chapter 4. Results

Note that Theorems A and B are incomparable in the sense that neither theorem implies the other. Furthermore, graphs satisfying the conditions of Theorem A or B need not be cycle extendable, vertex pancyclic or even pancyclic (see Remarks 4.1.2, 4.2.1, 4.2.5, 4.2.6, and 4.2.10). We will show, however, that every graph satisfying the conditions of one of these theorems has strong cyclic properties. This investigation is inspired by [9], in which Asratian and Sarkisian study cyclic properties of graphs satisfying the conditions of Theorem 3.1.7.

The main results of the thesis are Theorems 4.2.1, 4.2.3, 4.2.5, and 4.2.7 and Theorems 4.3.1, 4.3.2, 4.3.5, and 4.3.6. The results in Section 4.2 extend Theorems A and B and generalize Theorem 3.2.5, and the related results in Section 4.3 give a corollary that strengthens Theorem 3.2.8. In Section 4.4 we characterize all bipartite graphs that satisfy the conditions of Theorems A and B.

4.1 Preliminary results

We begin by stating some properties of the conditions in question. (A.1) For any path uwv with uv /∈ E(G) we have

d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)| − 1.

(B.1) For any path uwv with uv /∈ E(G) we have

d(u) + d(v) ≥ |M₂(w)| − 1.

(A.2) For every pair of vertices u, v with d(u, v) = 2 we have |N (u) ∩ N (v)| ≥ 2.

(B.2) Every 2-ball in G is 2-connected. Property 1. Condition (B.1) implies Condition (A.1).

Property 2. Condition (A.2) implies Condition (B.2) if G is connected and has at least three vertices.

Proof. If G is connected and has at least three vertices, then every 2-ball in G

has at least three vertices and is 2-connected if and only if it has no cut vertex. Assume that w is a cut vertex in a 2-ball. Then clearly w is either the center vertex of the ball, or adjacent to the center vertex. If w is the center of the 2-ball, pick two neighbours u and v of w in different components of G₂(w) − {w}. If w is not the center of the 2-ball, then denote by v the center and pick a vertex u in a component of G₂(v) − {w} that does not contain v. In both cases d(u, v) = 2 but |N (u) ∩ N (v)| = 1.

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4.1. Preliminary results 21

Figure 4.1: A nonpancyclic and a nonhamiltonian graph.

Property 3. Let uwv be a path in G with uv /∈ E(G). Then the inequality in Condition (A.1) is equivalent to |N (u) ∩ N (v)| ≥N (w) \ N (u) ∪ N (v)

− 1. Property 4. Let uwv be a path in G with uv /∈ E(G). Then the inequality in Condition (B.1) is equivalent to |N (u) ∩ N (v)| ≥M2(w) \ N (u) ∪ N (v)

− 1.

Remark 4.1.1. Conditions (A.1) and (B.2), the two weakest conditions, are not

enough to imply pancyclicity or even Hamiltonicity; two counterexamples are the graphs in Figure 4.1, which lack 4-cycles and Hamilton cycles, respectively.

Remark 4.1.2. The conditions do not imply pancyclicity without exceptions, as

the graph K_n,n satisfies all four conditions for n ≥ 2. Also, the graph C₅satisfies all conditions except (A.2).

Lemma 4.1.1. Every connected graph with at least three vertices satisfying Condition (B.2) is 2-connected.

Proof. Assume that G is not 2-connected, and let v be a cut vertex in G. Then v is a cut vertex of G₂(v), so G does not satisfy Condition (B.2).

Lemma 4.1.2. Every connected graph with at least three vertices satisfying Condition (A.2) is 2-connected.

Proof. This follows from Lemma 4.1.1 and Property 2.

Lemma 4.1.3. Let G be a graph satisfying the conditions of Theorem A. Then every edge of G lies on a cycle of length at most 4.

Proof. Let e = vw be an arbitrary edge in G. Since G is 2-connected and has at

least three vertices, d(w) ≥ 2. Let u be another neighbour of w. If e does not lie on a triangle then d(u, v) = 2. Now |N (u) ∩ N (v)| ≥ 2 by Condition (A.2), so e lies on a 4-cycle. The lemma follows.

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Lemma 4.1.4. Let G be a graph satisfying the conditions of Theorem A. Then every vertex of G lies on a cycle of length at most 4.

Proof. Since G is connected and has at least three vertices, every vertex is the

endpoint of an edge. Thus this follows from Lemma 4.1.3.

Lemma 4.1.5. Let G be a graph satisfying the conditions of Theorem B, and let u, v, and w be vertices such that d(u, v) = 2 and N (u) ∩ N (v) = {w}. Then the path uwv lies on a 5-cycle.

Proof. Clearly the path uwv cannot lie on any cycle of length shorter than 5.

Assume that uwv does not lie on a 5-cycle, and consider the ball G₂(w). Because

G₂(w) is 2-connected, there is a u–v-path P contained in G2(w) − {w}. Thus

V (P ) ⊂ M2(w). Since uwv does not lie on any cycle of length less than 6, it

follows that there is at least one vertex x in V (P ) \ {u, v} that is not adjacent to u or v. Thus the set V (P ) \ N (u) ∪ N (v) contains at least the three vertices

u, v, and x. But now, by Property 4,

1 = |N (u) ∩ N (v)| ≥M2(w) \ N (u) ∪ N (v) − 1 ≥ V (P ) \ N (u) ∪ N (v) − 1 ≥ 2, (4.1)

which is clearly a contradiction. Thus uwv lies on a 5-cycle.

Lemma 4.1.6. Let G be a graph satisfying the conditions of Theorem B. Then every edge of G lies on a cycle of length at most 5.

Proof. Let e = vw be an arbitrary edge in G. Since G is 2-connected and has

at least three vertices, d(w) ≥ 2. Let u be another neighbour of w. If e does not lie on a triangle then d(u, v) = 2, and if e does not lie on a 4-cycle then

N (u) ∩ N (v) = {w}. Now by Lemma 4.1.5, there is a 5-cycle containing e. The

lemma follows.

Lemma 4.1.7. Let G be a graph satisfying the conditions of Theorem B. Then every vertex of G lies on a cycle of length at most 5.

Proof. Since G is connected and has at least three vertices, every vertex is the

endpoint of an edge. Thus this follows from Lemma 4.1.6.

Remark 4.1.3. The limit of length 5 in Lemmas 4.1.6 and 4.1.7 cannot be

strengthened, as the graphs in Figure 4.2 satisfy the conditions of Theorem B but contain vertices and edges that do not lie on any cycle of length less than 5.

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4.2. Cycles through vertices 23

Figure 4.2: Two graphs with vertices and edges that do not lie on any cycle of length less than 5.

x

Figure 4.3: The graph K2∨ K3∨ K3∨ K1.

4.2 Cycles through vertices

Graphs satisfying the conditions of Theorem A

We will prove the following extension of Theorem A, which is a generalization of Theorem 3.2.5:

Theorem 4.2.1. Let G be a graph satisfying the conditions of Theorem A. Then for every nonhamiltonian cycle C_n of length n in G there is a cycle C_n+`of length n + `, where 1 ≤ ` ≤ 2, such that V (Cn) ⊂ V (Cn+`), unless n = |V (G)|− 1

and G ∈ K.

Remark 4.2.1. Graphs satisfying the conditions of Theorem A need not be cycle

extendable, even when excluding bipartite graphs. Consider for example the graphs K₂∨ K_n∨ K_n∨ K₁ for n ≥ 2 (see Figure 4.3). The graphs satisfy the conditions of Theorem A, but no 4-cycle containing the unique vertex x of degree n can be expanded to a 5-cycle.

Remark 4.2.2. The condition |N (u) ∩ N (v)| ≥ 2 for any vertices u and v with d(u, v) = 2 in Theorem 4.2.1 cannot be weakened to the condition that every

2-ball is 2-connected. Let G₁ be a complete graph with vertices {u₁, . . . , u_n}, let

G₂be an empty graph with vertices {v₁, . . . , v_n+1}, let G₃be the path v₁xyv_n+1, and let G = (G₁∨ G₂) ∪ G₃, as seen in Figure 4.4. The graph G satisfies the weaker conditions, but the 2n-cycle v₁u₁v₂u₂· · · u_nv₁ cannot be extended by less than three vertices.

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u₁ u_n

v₁ v₂ · · · v_n+1

x y

Figure 4.4: A graph with a cycle that cannot be expanded by less than three vertices.

Corollary 4.2.2. Let G be a bipartite graph satisfying the conditions of Theo-rem A. Then G is bi-cycle extendable unless G = Kn,n+1 for some n ≥ 2.

Lemma 4.1.4 tells us that every vertex in a graph satisfying the conditions of Theorem A lies on a cycle of length at most 4, which means that we can reformulate Theorem 4.2.1 in the following way:

Theorem 4.2.3. Let G be a graph that satisfies the conditions of Theorem A. Then for each vertex x ∈ V (G) there is a number r and a sequence of integers

n₁, n₂, . . . , n_r, depending on x, such that n1≤ 4, nr= |V (G)| (unless G ∈ K, in

which case n_r= |V (G)| − 1), and 1 ≤ n_i+1− n_i ≤ 2 for each i = 1, . . . , r − 1, and a sequence of cycles C_n

1, Cn2, . . . , Cnr of lengths n1, n2, . . . , nrrespectively,

such that x ∈ V (C_n₁) ⊂ V (C_n₂) ⊂ · · · ⊂ V (C_nr).

Remark 4.2.3. The sequence of cycles in Theorem 4.2.3 can be chosen such that

it contains at least two cycles unless G is one of the graphs C₃, C₄, and K_2,3.

Remark 4.2.4. If G is regular or |V (G)| is even, then G /∈ K, so n_r= |V (G)| for every vertex v in Theorem 4.2.3.

Remark 4.2.5. Graphs satisfying the conditions of Theorem A need not be vertex

pancyclic, even when excluding bipartite graphs. Consider for example the graphs

K₂∨ K_n∨ K_n∨ K₁ for n ≥ 2 (see Figure 4.3). The graphs satisfy the conditions of Theorem A, but the unique vertex x of degree n does not lie on any 3-cycle or 5-cycle.

Corollary 4.2.4. Let G be a bipartite graph satisfying the conditions of Theo-rem A. Then G is vertex bipancyclic unless G = Kn,n+1 for some n ≥ 2.

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Proof of Theorem 4.2.1. Assume that there is no cycle of length n + 1 or n + 2

containing the vertices of C_n. Specify a cyclic orientation of C_nand pick a vertex

v ∈ V (G) \ V (Cn) such that N (v) ∩ V (Cn) 6= ∅. Set W = N (v) ∩ V (Cn) and p = |W |. Let w₁, . . . , w_p be the vertices of W, occurring on ~C_n in the order of their indices, and set W+= {w1+, . . . , w

+

p}. All indices are considered to be

modulo p, so w_p+1= w₁.

Claim 1. The set W+∪ {v} is independent, N (w+

i ) ∩ N (v) = N (w + i ) ∩ W, |N (wi) ∩ W+| = |N (wi+) ∩ W |, and N (wi) \ N (w+i ) ∪ N (v) ∪ {v} ⊆ W+ for i = 1, . . . , p.

Proof. If there is an edge vw+_i ∈ E(G), then G contains an (n + 1)-cycle

w_ivw+_i C~_nw_i, and if there is an edge w_i+w_j+∈ E(G), then G contains an (n + 1)-cycle w_ivw_jC ~ nw + i w + jC~nwi. Thus W+∪ {v} is an independent set. (4.2) Also, if N (w+i ) ∩ N (v) \ V (Cn) 6= ∅ for some 1 ≤ i ≤ p, that is, if w

+

i and v have a common neighbour u outside C_n, then G contains an (n + 2)-cycle

w_ivuw_i+C~_nw_i. Thus N (w_i+) ∩ N (v) \ V (Cn) = ∅, which means that

N (w_i+) ∩ N (v) = N (w_i+) ∩ W. (4.3) Now for each i = 1, . . . , p, we have d(v, wi+) = 2 and wi ∈ N (w+i ) ∩ N (v), so

by Property 3, |N (w+ i ) ∩ W | = |N (w + i ) ∩ N (v)| ≥ N (w_i) \ N (w+_i ) ∪ N (v) − 1. (4.4) Obviously, N (w_i) ∩ W+⊆ N (w_i) \ N (w_i+) ∪ N (v) ∪ {v}. (4.5) Thus |N (w_i) ∩ W+| ≤ N (w_i) \ N (w+_i ) ∪ N (v)

− 1. This and (4.4) together imply that

|N (wi) ∩ W

+

| ≤ |N (w+i ) ∩ W |. (4.6)

We will now count the number of edges between W+ and W in two different ways: e(W+, W ) = p X i=1 |N (wi) ∩ W + | ≤ p X i=1 |N (wi+) ∩ W | = e(W + , W ). (4.7)

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It follows for each i = 1, . . . , p, that

|N (w_i) ∩ W+| = |N (w+

i ) ∩ W | (4.8)

and that we have equality in (4.5), so

N (w_i) \ N (w+_i ) ∪ N (v) ∪ {v} = N (wi) ∩ W

+

⊆ W+. (4.9)

Claim 2. w_i+= w−_i+1 for i = 1, . . . , p, that is, n = 2p and v is adjacent to every second vertex of Cn.

Proof. Suppose that v is not adjacent to every second vertex of the cycle C_n. Then w_i+6= w_i+1− for some i. Without loss of generality, assume that w+₁ 6= w−₂, which means that w−₂ ∈ W/ +_{. This and (4.9) for i = 2 imply that w}−

2 ∈ N (w + 2),

because otherwise w−2 ∈ N (w2) \ N (w2+) ∪ N (v) ∪ {v} ⊆ W+, a contradiction.

Therefore w−₂w+₂ ∈ E(G). This in turn means that w+ 2 6= w

−

3, because otherwise

there would be an (n + 1)-cycle w2−w +

2w2vw3C~nw−2 (unless p = 1, in which case

recall that w_p+1= w₁and skip this sentence). Repetition of this argument shows that w+_i 6= w−

i+1 for i = 1, . . . , p, and that

w+_i w−_i ∈ E(G) for each i = 1, . . . , p. (4.10) Now it is easy to see that w₁+w_j ∈ E(G) for each j 6= 1, as otherwise there/

would be an (n + 1)-cycle w₁vw_jw+₁C~_nw_j−w+_jC~_nw₁ containing the vertices of C_n. This, together with (4.3), implies that N (w+₁)∩N (v) = {w₁}. Since d(w+

1, v) = 2,

this contradicts Condition (A.2). Thus we can conclude that w+i = wi+1− for each i = 1, . . . , p, and that n = 2p.

Claim 3. n = |V (G)| − 1 and G ∈ K.

Proof. We have concluded that n = 2p and that N (v) contains every second

vertex of C_n. Note that this means that p ≥ 2. If V (C_n) ∪ {v} 6= V (G), consider a vertex u ∈ V (G)\ V (Cn)∪{v} with N (u)∩V (Cn) 6= ∅ (such a vertex must exist,

since G is 2-connected by Lemma 4.1.2). Since v was picked arbitrarily in the set

V (G) \ V (C_n) such that N (v) ∩ V (Cn) 6= ∅, we can conclude that u is adjacent

to every second vertex of C_n as well. Now suppose that N (u) ∩ V (C_n) = W+=

V (C_n) \ N (v). Then there is an (n + 2)-cycle w₁vw₂w₁+uw+₂C~_nw₁containing the vertices of C_n, a contradiction. Therefore N (u) ∩ V (C_n) = W. Thus all vertices outside C_n that have neighbours in C_n are adjacent to all vertices in W, and no vertex outside Cn is adjacent to any vertex in W

+

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Since W+is independent, it now follows that N (w_i+) ⊆ W for all i = 1, . . . , p. Thus (4.8) means that |N (w_i) ∩ W+| = |N (w+

i ) ∩ W | = d(w

+

i ). The reasoning

that leads up to (4.8) can also be used to show that |N (wi) ∩ W+| = d(wi−).

Since w_i+= w_i+1− for all i = 1, . . . , p, these equalities mean that there is some number t such that

|N (w₁) ∩ W+| = d(w+ 1) = |N (w2) ∩ W +_{| = · · · = d(w}+ p) = t. (4.11) Now clearly N (w_i) ∩ W+ ∪ N (w+ i ) ∪ N (w − i ) \ N (w + i ) ⊆ N (w_i) \ {v} ∪ N (w+ i ) ∪ N (w − i ), (4.12)

and since the left hand side is a disjoint union, we can use (4.11) along with Condition (A.1) to get the following:

2t + |N (w_i−) \ N (w+_i )| = N (w_i) ∩ W+ ∪ N (w+_i ) ∪ N (w−_i ) \ N (w_i+) ≤ N (w_i) \ {v} ∪ N (w+_i ) ∪ N (w−_i ) = |N (wi−) ∪ N (wi+) ∪ N (wi)| − 1 ≤ d(wi+) + d(w−i ) = 2t. (4.13)

This means that N (w−i ) \ N (w+i ) = ∅, so N (w−i ) ⊆ N (w+i ) for all i = 1, . . . , p,

and since d(w−_i ) = d(w_i+) this implies that N (w_i−) = N (w+_i ) for all i. Thus

N (w₁+) = N (w2+) = · · · = N (w +

p). Since wi∈ N (w

+

i ) for every i, it follows that

every vertex in W+ is adjacent to all vertices in W, so t = p.

Finally we will show that G contains no vertices except v outside C_n, which means that G ∈ K and n = |V (G)| − 1. If there is a such a vertex, then there must exist a vertex u outside V (C_n) ∪ {v} that has a neighbour in C_n, because

G is 2-connected. As we saw earlier, this means that N (u) ∩ V (Cn) = W, so u ∈ N (w₁). But then V (C_n) ∪ {v, u} ⊆ N (w−₁) ∪ N (w₁) ∪ N (w₁+), so

2p = d(w+₁) + d(w₁−) ≥ |N (w−₁) ∪ N (w₁) ∪ N (w₁+)| − 1 ≥ 2p + 1, (4.14) which is a contradiction. Thus V (G) \ V (Cn) ∪ {v}

does not contain any vertices, so G ∈ K and n = |V (G)| − 1. The proof of Claim 3 is completed. The theorem now follows.

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Graphs satisfying the conditions of Theorem B

We will prove the following extension of Theorem B, which is a generalization of Theorem 3.2.5:

Theorem 4.2.5. Let G be a graph satisfying the conditions of Theorem B. Then for every nonhamiltonian cycle C_n of length n in G there is a cycle C_n+`of length n + `, where 1 ≤ ` ≤ 2, such that V (C_n) ⊂ V (C_n+`), unless n = |V (G)|− 1 and G ∈ K.

Remark 4.2.6. Graphs satisfying the conditions of Theorem B need not be cycle

extendable, even when excluding bipartite graphs and the graph C₅. Consider for example the graphs K₂∨ K_n∨ K_n∨ K₁for n ≥ 2 (see Figure 4.3 on page 23). The graphs satisfy the conditions of Theorem B, but no 4-cycle containing the unique vertex x of degree n can be expanded to a 5-cycle.

Remark 4.2.7. The condition d(u) + d(v) ≥ |M₂(w)| − 1 for every path uwv with

uv /∈ E(G) in Theorem 4.2.5 cannot be weakened to the condition d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)| − 1. Let G₁be a complete graph with vertices {u₁, . . . , u_n}, let G₂ be an empty graph with vertices {v₁, . . . , v_n+1}, let G₃ be the path

v1xyvn+1, and let G = (G1∨ G2) ∪ G3, as seen in Figure 4.4 on page 24. The

graph G satisfies the weaker conditions, but the 2n-cycle v₁u₁v₂u₂· · · u_nv₁cannot be extended by less than three vertices.

Corollary 4.2.6. Let G be a bipartite graph satisfying the conditions of Theo-rem B. Then G is bi-cycle extendable unless G = K_n,n+1 for some n ≥ 2.

Lemma 4.1.7 tells us that every vertex in a graph satisfying the conditions of Theorem B lies on a cycle of length at most 5, which means that we can reformulate Theorem 4.2.5 in the following way:

Theorem 4.2.7. Let G be a graph that satisfies the conditions of Theorem B. Then for each vertex x ∈ V (G) there is a number r and a sequence of integers

n₁, n₂, . . . , n_r, depending on x, such that n₁≤ 5, n_r= |V (G)| (unless G ∈ K, in which case nr= |V (G)| − 1), and 1 ≤ ni+1− ni ≤ 2 for each i = 1, . . . , r − 1,

and a sequence of cycles C_n₁, C_n₂, . . . , C_nr of lengths n₁, n₂, . . . , n_rrespectively, such that x ∈ V (C_n

1) ⊂ V (Cn2) ⊂ · · · ⊂ V (Cnr).

Remark 4.2.8. The sequence of cycles in Theorem 4.2.7 can be chosen such that

it contains at least two cycles unless G is one of the graphs C₃, C₄, C₅, and K_2,3.

Remark 4.2.9. If G is regular or |V (G)| is even, then G /∈ K, so n_r= |V (G)| for every vertex v in Theorem 4.2.7.

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Remark 4.2.10. Graphs satisfying the conditions of Theorem B need not be vertex

pancyclic, even when excluding bipartite graphs and the graph C₅. Consider for example the graphs K₂∨ Kn∨ Kn∨ K1for n ≥ 2 (see Figure 4.3 on page 23).

The graphs satisfy the conditions of Theorem B, but the unique vertex x of degree n does not lie on any 3-cycle or 5-cycle.

Corollary 4.2.8. Let G be a bipartite graph satisfying the conditions of Theo-rem B. Then G is vertex bipancyclic unless G = K_n,n+1 for some n ≥ 2.

We give the proof of Theorem 4.2.5. Note that the proofs of Claim 1 in Theorems 4.2.1 and 4.2.5 are almost identical, and the proofs of Claim 3 are very similar as well.

Proof of Theorem 4.2.5. Assume that there is no cycle of length n + 1 or n + 2

containing the vertices of C_n. Specify a cyclic orientation of C_nand pick a vertex

v ∈ V (G) \ V (C_n) such that N (v) ∩ V (C_n) 6= ∅. Set W = N (v) ∩ V (C_n) and

p = |W |. Let w₁, . . . , w_p be the vertices of W, occurring on ~C_n in the order of their indices, and set W+= {w1+, . . . , wp+}. All indices are considered to be

modulo p, so w_p+1= w₁.

Claim 1. The set W+∪ {v} is independent, N (w+

i ) ∩ N (v) = N (w + i ) ∩ W, |N (w_i) ∩ W+| = |N (w+ i ) ∩ W |, and M2(wi) \ N (w + i ) ∪ N (v) ∪ {v} ⊆ W + for i = 1, . . . , p.

Proof. If there is an edge vw+i ∈ E(G), then G contains an (n + 1)-cycle w_ivw+_i C~_nw_i, and if there is an edge w_i+w_j+∈ E(G), then G contains an (n + 1)-cycle w_ivw_jC ~ nw + i w + jC~nwi. Thus W+∪ {v} is an independent set. (4.15) Also, if N (w+_i ) ∩ N (v) \ V (Cn) 6= ∅ for some 1 ≤ i ≤ p, that is, if w

+

i and v have a common neighbour u outside C_n, then G contains an (n + 2)-cycle

w_ivuw_i+C~_nw_i. Thus N (w_i+) ∩ N (v) \ V (Cn) = ∅, which means that

N (w_i+) ∩ N (v) = N (w_i+) ∩ W. (4.16) Now for each i = 1, . . . , p, we have d(v, w_i+) = 2 and w_i ∈ N (w+

i ) ∩ N (v), so by Property 4, |N (w+i ) ∩ W | = |N (w + i ) ∩ N (v)| ≥ M2(wi) \ N (w + i ) ∪ N (v) − 1. (4.17)

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Obviously,

N (wi) ∩ W+⊆ M2(wi) \ N (w+i ) ∪ N (v) ∪ {v}. (4.18)

Thus |N (w_i) ∩ W+| ≤M₂(w_i) \ N (w+_i ) ∪ N (v)

− 1. This and (4.17) together imply that

|N (wi) ∩ W+| ≤ |N (w+i ) ∩ W |. (4.19)

We will now count the number of edges between W+ and W in two different ways: e(W+, W ) = p X i=1 |N (wi) ∩ W + | ≤ p X i=1 |N (wi+) ∩ W | = e(W + , W ). (4.20)

It follows for each i = 1, . . . , p, that

|N (w_i) ∩ W+| = |N (w+

i ) ∩ W | (4.21)

and that we have equality in (4.18), so

M₂(w_i) \ N (w_i+) ∪ N (v) ∪ {v} = N (w_i) ∩ W+⊆ W+_. _(4.22)

Claim 2. w_i+= w−_i+1 for i = 1, . . . , p, that is, n = 2p and v is adjacent to every second vertex of C_n.

Proof. Suppose that v is not adjacent to every second vertex of the cycle C_n. Then w_i+6= w−

i+1 for some i. Without loss of generality, assume that w

+ 1 6= w−2,

which means that w−₂ ∈ W/ +_{. This and (4.22) for i = 2 imply that w}− 2 ∈ N (w

+ 2),

because otherwise w₂−∈ M₂(w₂) \ N (w+₂) ∪ N (v) ∪ {v} ⊆ W+_{, a contradiction.}

Therefore w−2w +

2 ∈ E(G). This in turn means that w +

2 6= w3−, because otherwise

there would be an (n + 1)-cycle w₂−w₂+w₂vw₃C~_nw−₂ (unless p = 1, in which case recall that wp+1= w1and skip this sentence). Repetition of this argument shows

that w+_i 6= w−_i+1 for i = 1, . . . , p, and that

w+_i w−_i ∈ E(G) for each i = 1, . . . , p. (4.23) Now it is easy to see that wi+wj ∈ E(G) for each i = 1, . . . , p and each j 6= i,/

as otherwise there would be an (n + 1)-cycle w_ivw_jw_i+C~_nw_j−w_j+C~_nw_i containing the vertices of C_n. Thus, by (4.16),

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The same holds true for w−_i, so

N (w_i−) ∩ N (v) = {w_i}. (4.25) Clearly {w+ i , v} ⊆ M2(wi) \ N (w + i ) ∪ N (v), (4.26) so, by Property 4, 1 = |N (w+_i ) ∩ N (v)| ≥M₂(w_i) \ N (w+_i ) ∪ N (v) − 1 ≥ 1. (4.27) We have equality in (4.27) as well as in (4.26), which means that

M₂(wi) ⊆ N (w

+

i ) ∪ N (v) ∪ {w

+

i , v}. (4.28)

The same argument can again be used with w−i instead of w

+

i. Since clearly N (w_i−) ∪ {w−_i } ⊂ M₂(w_i), we can use (4.25) and (4.28) together with the fact that v /∈ N (wi−) (otherwise there would be an (n + 1)-cycle wi−vwiC~nw−i ) to

show that N (w_i−) ∪ {w−_i } ⊆ N (w+ i ) ∪ {w + i }. Switching w + i and w − i, the same

argument can be used to prove the reverse inclusion, so

N (w+_i ) ∪ {w_i+} = N (w− i ) ∪ {wi−}, (4.29) and in particular w−−_i ∈ N (w+ i ) and w ++ i ∈ N (w − i ). (4.30)

Now by Lemma 4.1.5 there is a cycle vw₁w₁+abv for some vertices a and b.

We shall see that neither a nor b lies on Cn. If b lies on Cn then b ∈ W, so the

equalities above are valid for b+ and b−. Then a ∈ V (C_n), because otherwise there would be an (n + 2)-cycle w₁vbaw₁+C~_nb−b+C~_nw₁. Now if a+ ∈ N (v), which would mean that a = w−_i for some i, then N (a) ∩ N (v) = {a+} by (4.25), which is a contradiction as that set must contain b (note that a 6= b−, because if a = b− then b+ is adjacent to w1+ by (4.29), contradicting the fact

that W+ is independent). Thus a+∈ M₂(b) \ N (v) ∪ {b+, v}, because clearly

a+ ∈ M2(b) \ {b

+_{, v}. From (4.28) we then get a}+

∈ N (b+), so there would be an (n + 1)-cycle w₁−w₁++C~_naw₁+w₁vbC ~ na +_b+_C_~ nw − 1 (assuming a ∈ w1C~nb; if a ∈ b ~C_nw₁a similar (n + 1)-cycle can be constructed). Thus b /∈ V (C_n).

If a ∈ V (C_n) then N (b) ∩ V (C_n) 6= ∅. Then considering b instead of v, we can repeat the reasoning in the proof of Claim 1 and the first paragraph of the proof of Claim 2 to show that either N (b) contains every second vertex of Cn,

Some cyclic properties of graphs with local Ore-type conditions