# (a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from

Save this PDF as:

Size: px
Start display at page:

Download "(a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from"

## Transcription

1 4 Graph Theory Throughout these notes, a graph G is a pair (V, E) where V is a set and E is a set of unordered pairs of elements of V. The elements of V are called vertices and the elements of E are called edges. We typically denoted by V (G) = V the vertex set of G and E(G) = E the edge set of G. If u, v V (G), then u and v are adjacent if {u, v} E(G). We refer to u and v as the endpoints of the edge {u, v}. It is convenient to draw V (G) as a set of points in the plane and each edge as a curve between its endpoints in the plane, as in Figure 1(a). A multigraph is a pair (V, E) where V is a set and E is a multiset of unordered pairs of elements of V we allow E to contain repetitions of the same unordered pair. In other words, more than one edge can join two vertices. Sometimes we even allow loops on a vertex see Figure 1(b). A directed graph digraph for short is a pair (V, E) where V is a set and E is a set of ordered pairs of G. The elements of E are called arcs, and we draw them in the plane as curves with an arrow indicating the direction of the arc see Figure 1(c). (a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from the topmost vertex, then the edge set of that graph is {{1, 2}, {2, 3}, {3, 4}, {4, 2}}. Under the same labelling, the multigraph in Figure 1(b) would have edge multiset {{1, 2}, {2, 3}, {3, 4}, {4, 2}, {4, 2}, {3, 3}}. Finally, the digraph in Figure 1(c) would have arc set {(2, 1), (2, 3), (3, 4), (4, 2)}. 1

2 4.1 Graph theoretic terminology In the next few short sections, we describe the notation that will be used in the remainder of the notes to refer to various properties of graphs Degrees and Neighbourhoods The neighborhood of v in G is denoted Γ G (v), and is the set of vertices of G which are adjacent to v. The degree of a vertex v in a graph G is the number of edges incident with v, and is denoted d G (v), or just d(v) for short. We write δ(g) and (G) for the smallest and largest degree of a vertex in G, respectively. For example, if G is the graph in Figure 1(a), then δ(g) = 1 and (G) = 3. The neighborhood of a set X V (G), which we denote by Γ(X), is the set of vertices of V (G)\X with at least one neighbor in X. For sets X, Y V (G), we write e(x, Y ) for the set of edges of G with one endpoint in X and the other endpoint in Y. An important fact involving the degrees of a graph G, which we will use on numerous occasions, is the handshaking lemma: Lemma 1 For any graph G = (V, E), d G (v) = 2 E. v V Proof When we add up the degrees of vertices of G, every edge of G is counted twice, once for each of its endpoints. An important consequence of the handshaking lemma is that the number of vertices of odd degree in any graph must be even otherwise the sum on the left above would be odd. So, for example, if you try to draw a graph on five vertices in which every vertex has degree three, you would fail. We refer to graphs where all the degrees are the same, say all degrees are equal to r, as r-regular graphs. The handshaking lemma gives an easy way to count the number of edges in a graph: it is just half the sum of the degrees of the vertices Basic classes of graphs Let K n denote the complete graph on n vertices: this is the graph consisting of all ( ) n 2 possible edges on n vertices. A bipartite graph is a graph G such that V (G) can be partitioned into two sets A and B such that each edge of G has one endpoint in A and one endpoint in B. The sets A and B are called the parts of G. Let K s,t denote 2

3 the complete bipartite graph with s vertices in one part and t vertices in the other: this graph consists of all st possible edges between a set of size s and a set of size t. We refer to K 1,t as a star. The n-dimensional cube, or n-cube, is the graph whose vertex set is the set of binary strings of length n, and whose edge set consists of pairs of strings differing in one position. A tree is a connected graph without cycles, and a forest is a vertex-disjoint union of trees and isolated vertices. The vertices of degree one in a tree or forest are often called leaves. A cycle is a connected graph all of whose vertices have degree two, and a path is a tree with exactly two vertices of degree one. Those vertices are sometimes called the endpoints of the paths, and the other vertices of the path are called internal vertices. Two paths are internally disjoint if they share no internal vertices. The number of edges in K n is ( n 2), and by the handshaking lemma, we can verify this: every vertex in K n is joined to all n 1 other vertices, so every vertex has degree n 1. By the handshaking lemma E(K n ) = 1 d(v) = 1 (n 1) = 1 ( ) n n(n 1) =. 2 v V v V For the n-dimensional cube, Q n, there are 2 n vertices. Each vertex v is joined to n other vertices namely flip one position in the string v to get strings adjacent to v, and there are n possible positions to do a flip. So every vertex of the n-cube has degree n, and E(Q n ) = 1 d(v) = 1 n = n n = n2 n 1. v V v V Subgraphs If H and G are graphs and V (H) V (G) and E(H) E(G), then H is called a subgraph of G. To denote that H is a subgraph of G, we write H G. If in addition V (H) = V (G) then H is called a spanning subgraph of G. If G = (V, E) is a graph and X V and L E, then G X denotes the graph whose vertex set is V \X and whose edge set consists of all edges disjoint from X. Also G L denotes the graph (V, E\L). In the case that X = {x}, we write G x instead of G X and if L = {e} we write G e instead of G {e}. For example, if G is the graph in Figure 1(a) and x is the vertex of degree three in Figure 1(a), and y is the vertex of degree one, then G x consists of a disjoint union of K 1 and K 2, and G y is K 3. 3

4 4.1.4 Walks and Paths A walk in a graph G = (V, E) is an alternating sequence of vertices and edges, whose first and last elements are vertices, and such that each edge joins the vertices immediately preceding it and succeeding it in the sequence. For example, a{a, d}d{d, e}e{e, a}a{a, d}d is a walk in the graph in Figure 3. If the vertices of a walk are all distinct, then it is referred to as a path. The length of a walk is the number of steps taken in the walk. The first and last vertices of a walk are called end vertices and the remaining vertices are called internal vertices. For example, the walk given above is not a path, but a{a, d}d{d, e}e is a path. The walk above has length four. For convenience, we often omit the edges and list the vertices in order along the path, such as ade instead of a{a, d}d{d, e}e. If the endpoints of a walk are u and v, then we say the walk is a uv-walk. If the first and last vertex of the walk are the same, then the walk is referred to as a closed walk. A cycle is a closed walk with the same number of edges as vertices every vertex of a cycle has degree two. For example, in Figure 3, the different cycles are adea, abea, bcdeb, adeba, adcba, aedcba, and aebcda. d c e a b Figure 3 : Walks, paths and cycles. 4.2 Connected graphs A graph is connected if any pair of vertices in the graph are joined by at least one path. If a graph is not connected, we say it is disconnected. The components of a graph G = (V, E) are the maximal (i.e. with as many edges as possible) connected subgraphs of G. The graphs in Figure 1(a) and Figure 3 are connected, whereas the graph below has three components. 4

5 Figure 4 : Components. The simplest type of connected graph is a tree: by definition, a tree is a connected graph without cycles a connected acyclic graph. To describe the structure of trees, we define the notion of a bridge. A bridge of a graph G is an edge e E(G) such that G e has more components than G. For example, in Figure 1(a), the top edge e is a bridge since G has one component but G e has two components. It is easy to spot the bridges of a graph, using the following lemma: Lemma 2 An edge e E(G) is a bridge if and only if e is not contained in any cycle in G. Proof If e is contained in a cycle C of G, then C e is a path joining the ends of e. But that means G e is connected, so e could not have been a bridge. Since a tree has no cycles, every edge of a tree must be a bridge. We can now characterize which graphs are trees in a few ways. Proposition 3 Each of the following is equivalent to a graph G being a tree: 1. The graph G is connected and acyclic. 2. The graph G is connected and every edge of G is a bridge. 3. The graph G is connected and has V (G) 1 edges. Proof Clearly Proposition 3.1 is the definition of G being a tree. Since a connected graph is acyclic if and only if every edge of the graph is a bridge, by the last lemma, Proposition 3.1 and Proposition 3.2 are equivalent. We proved Proposition 3.1 implies Proposition 3.3 by strong induction on the number of vertices in the tree, so it remains to show that Proposition 3.3 implies Proposition 3.1. To see that, if G is connected with V (G) 1 edges, we remove an edge of any cycle and that does not disconnect G, by Lemma 2. We continue removing edges of G in cycles until all the cycles are gone. But then the remaining graph T is connected and acyclic, so must be a tree. 5

6 Since it has V (G) vertices, we know it must have V (G) 1 edges. But G itself has V (G) 1 edges, so G = T. The last part of the proof of this proposition is important. It says that in any connected graph G, while there is a cycle, pick an edge of the cycle and remove it. By Lemma 2, we did not disconnect the graph, so if we repeat this procedure we eventually obtain a spanning subgraph of G which is acyclic and connected a tree. We call this a spanning tree of the graph. In general, a graph has many spanning trees. Proposition 4 Any connected graph contains a spanning tree. The proof gives a fairly quick way to find a spanning tree of a graph: search for a cycle and remove an edge of the cycle, and repeat until there are no cycles left. 4.3 Block Decomposition We just gave three equivalent characterizations of trees. In general, we would like to describe how to build connected graphs. The main result will be the block decomposition theorem. We require some definitions. A cut vertex of a graph G is a vertex G such that G x is disconnected. A block of a graph is a maximal connected subgraph with no cut vertex a subgraph with as many edges as possible and no cut vertex. So a block is either K 2 or is a graph which contains a cycle. For example in a tree, every block is K 2. The block decomposition of a graph is just the set of all the blocks of the graph. An example of a block decomposition is shown below. Figure 5 : Blocks. 6

7 In the picture, there are fourteen blocks. Seven of the blocks are K 2, four of the blocks are triangles, one of the blocks is K 5, and there are two other blocks. The block decomposition theorem says that block decompositions of graphs have a treelike structure. To make this precise, given a graph G, we form a new graph B where the vertices of B consist of all cut vertices of G and also all blocks of G, and where a block is joined to all cut vertices of G contained in it. An example of this graph is shown below for the figure above: Figure 6 : The graph B. In the figure, the black vertices represent cut vertices of G, and the grey vertices represent blocks of G. Here is the block decomposition theorem: Theorem 5 Let G be a connected graph. Then B is a tree. Proof By adding edges inside the blocks of G, we do not change B, so we can assume every block of G is a complete graph. Since G is connected, clearly B is connected too. Now we show B has no cycles. The vertices of a cycle C B are alternately blocks of G and cut vertices of G, by definition of B. This is shown in the figure below: 7

8 B v B B v Figure 7 : Cycle in B. In the figure, the blocks are shown as grey dots and labelled B and the cut vertices are black dots labelled v. Let the cut vertices of G in order along C be v 0, v 1,..., v k, v 0. Then v 0 v 1 v 2... v k v 0 is a cycle C G. If B C, then B C is a subgraph of G consisting of the complete graph B together with the cycle C containing an edge of B and at least one edge not in B. Therefore B C has no cut vertex, and must be a block of G. However, this contradicts the definition that B is block. Using this lemma, we give a first example of a structure theorem in graph theory. Define a theta graph to be any graph consisting of the union of three internally disjoint paths between two points. Proposition 6 Let G be a connected graph containing no theta graph. Then every block of G is a cycle or K 2 and G is a tree of cycles and K 2 s, as shown in Figure 8 below. Proof Let B be a block of G. If B K 2, then B contains a cycle, C. If B C, then there is a path P in B such that P C is a theta graph. Therefore B = K 2 or B is a cycle. We know by the last result that G is then a tree of cycles and K 2 s. 8

9 Figure 8 : Tree of cycles and K 2 s 4.4 k-connected graphs A cut of a connected graph G is a set of vertices or edges whose removal from G gives a disconnected graph. A graph G is k-connected if every cut of G has size at least k, and k-edge-connected if every edge cut of G has size at least k. So far we have characterized trees and connected graphs using block decompositions. The situation for k-connected graphs is more complicated when k 2. Fortunately, Menger s Theorems give a necessary and sufficient condition for a graph to be k-connected or k-edge-connected. Let u and v be vertices in a graph G, and let P and Q be uv-paths. Then P and Q are internally disjoint if the only vertices they have in common are u and v. A uv-separator is a set W V (G)\{u, v} such that u and v are in distinct components of G W. Finally, let κ(u, v) denote the minimum size of a uv-separator in G. Theorem 7 (Menger s Theorem Vertex Form) Let u and v be non-adjacent vertices in a graph G. Then the maximum number of pairwise vertex disjoint uv-paths in G is κ(u, v). In particular, a graph is k-connected if and only if each pair of its vertices is connected by at least k pairwise internally disjoint paths. It is important to note that condition that u and v are not adjacent in this theorem. Clearly, there is no uv-separator if u and v are adjacent. The vertex-form of Menger s Theorem has an edge-form. A set L E(G) is a uv-edge-separator if u and v are in different components of G L. Let λ(u, v) denote the minimum size of a uv-edgeseparator in G. 9

10 Theorem 8 (Menger s Theorem Edge Form) Let u and v be any vertices in a graph G. Then λ(u, v) equals the maximum number of pairwise edge-disjoint uv-paths in G. In particular, a graph is l-edge-connected if and only if each pair of its vertices is connected by at least l pairwise edge-disjoint paths. We will see how these theorems follow readily from the max-flow min-cut theorem. Note that it is possible to prove these theorems directly by structural arguments, but this is beyond the scope of this course. It is an interesting consequence of Menger s Theorem that if G is a graph with minimum degree δ(g), and κ(g) and λ(g) are the sizes of the smallest vertex cut and edge cut of G respectively, then κ(g) λ(g) δ(g). This is intuitively true in the sense that we can do more damage to a graph by removing vertices than edges, so we expect κ(g) λ(g). We prove this result as follows Corollary 9 Let G be any graph which is not a complete graph. Then κ(g) λ(g) δ(g). Proof To see that λ(g) δ(g), note that if we remove all the edges around a vertex of minimum degree, we disconnected G and therefore that set of δ(g) edges is an edge cut of G. Since λ(g) is the smallest possible size of an edge cut of G, we get λ(g) δ(g). To prove κ(g) λ(g), we use the two versions of Menger s Theorem given above. By definition, κ(g) = min{κ(u, v) : u, v V (G) are not adjacent}. Similarly λ(g) = min{λ(u, v) : u, v V (G)}. Now by Menger s Theorems, if u and v are not adjacent, then κ(u, v) λ(u, v) since a set of internally disjoint uv-paths is also a set of edge-disjoint uv-paths. If u and v are adjacent e = {u, v} E(G) then we remove the edge between them. Clearly that reduces λ(u, v) by exactly one, and also reduces the maximum number of edge-disjoint uv-paths by one. So we get κ(g e) λ(g) 1. However, κ(g e) κ(g) 1 so we get κ(g) 1 λ(g) 1 and the proof is complete: κ(g) λ(g). 10

### 1 Digraphs. Definition 1

1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania

Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### Planarity Planarity

Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?

### Graph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick

Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### Graph Theory. Judith Stumpp. 6. November 2013

Graph Theory Judith Stumpp 6. November 2013 ei dem folgenden Skript handelt es sich um einen Mitschrieb der Vorlesung Graph Theory vom Wintersemester 2011/2012. Sie wurde gehalten von Prof. Maria xenovich

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.

Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.

### Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest.

2 Trees What is a tree? Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest. Theorem 2.1 If G is a forest, then comp(g) = V (G) E(G). Proof: We proceed by induction on E(G).

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### Chapter 4: Trees. 2. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

9 Properties of Trees. Definitions: Chapter 4: Trees forest - a graph that contains no cycles tree - a connected forest. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### 3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

### Characterizations of Arboricity of Graphs

Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs

### 1. Relevant standard graph theory

Color identical pairs in 4-chromatic graphs Asbjørn Brændeland I argue that, given a 4-chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### Chapter 2. Basic Terminology and Preliminaries

Chapter 2 Basic Terminology and Preliminaries 6 Chapter 2. Basic Terminology and Preliminaries 7 2.1 Introduction This chapter is intended to provide all the fundamental terminology and notations which

### Chapter 6 Planarity. Section 6.1 Euler s Formula

Chapter 6 Planarity Section 6.1 Euler s Formula In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities

### Planar Graph and Trees

Dr. Nahid Sultana December 16, 2012 Tree Spanning Trees Minimum Spanning Trees Maps and Regions Eulers Formula Nonplanar graph Dual Maps and the Four Color Theorem Tree Spanning Trees Minimum Spanning

### CMSC 451: Graph Properties, DFS, BFS, etc.

CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs

### Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

### Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

- 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

### 14. Trees and Their Properties

14. Trees and Their Properties This is the first of a few articles in which we shall study a special and important family of graphs, called trees, discuss their properties and introduce some of their applications.

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### Social Media Mining. Graph Essentials

Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### DO NOT RE-DISTRIBUTE THIS SOLUTION FILE

Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1

GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept

### Introduction to Graph Theory

Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate

### Product irregularity strength of certain graphs

Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

### Graphs and Network Flows IE411 Lecture 1

Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture

### Lecture 4: The Chromatic Number

Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 4: The Chromatic Number Week 1 Mathcamp 2011 In our discussion of bipartite graphs, we mentioned that one way to classify bipartite graphs

### Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input

### GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### Homework 13 Solutions. X(n) = 3X(n 1) + 5X(n 2) : n 2 X(0) = 1 X(1) = 2. Solution: First we find the characteristic equation

Homework 13 Solutions PROBLEM ONE 1 Solve the recurrence relation X(n) = 3X(n 1) + 5X(n ) : n X(0) = 1 X(1) = Solution: First we find the characteristic equation which has roots r = 3r + 5 r 3r 5 = 0,

### Homework 15 Solutions

PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Chapter 2 Paths and Searching

Chapter 2 Paths and Searching Section 2.1 Distance Almost every day you face a problem: You must leave your home and go to school. If you are like me, you are usually a little late, so you want to take

### HOMEWORK #3 SOLUTIONS - MATH 3260

HOMEWORK #3 SOLUTIONS - MATH 3260 ASSIGNED: FEBRUARY 26, 2003 DUE: MARCH 12, 2003 AT 2:30PM (1) Show either that each of the following graphs are planar by drawing them in a way that the vertices do not

### CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

### 4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests

4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in

### Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing:

1. GRAPHS AND COLORINGS Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing: 3 vertices 3 edges 4 vertices 4 edges 4 vertices 6 edges A graph

### The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)

Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

### MGF 1107 CH 15 LECTURE NOTES Denson. Section 15.1

1 Section 15.1 Consider the house plan below. This graph represents the house. Consider the mail route below. This graph represents the mail route. 2 Definitions 1. Graph a structure that describes relationships.

### 1 Plane and Planar Graphs. Definition 1 A graph G(V,E) is called plane if

Plane and Planar Graphs Definition A graph G(V,E) is called plane if V is a set of points in the plane; E is a set of curves in the plane such that. every curve contains at most two vertices and these

### On Total Domination in Graphs

University of Houston - Downtown Senior Project - Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### Chapter 6: Graph Theory

Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Factors and Factorizations of Graphs

Factors and Factorizations of Graphs Jin Akiyama and Mikio Kano June 2007 Version 1.0 A list of some notation G denotes a graph. X Y X is a subset of Y. X Y X is a proper subset of Y. X Y X \ Y, where

### 6. GRAPH AND MAP COLOURING

6. GRPH ND MP COLOURING 6.1. Graph Colouring Imagine the task of designing a school timetable. If we ignore the complications of having to find rooms and teachers for the classes we could propose the following

### CHAPTER 2 GRAPHS F G C D

page 1 of Section 2.1 HPTR 2 GRPHS STION 1 INTROUTION basic terminology graph is a set of finitely many points called vertices which may be connected by edges. igs 1 3 show three assorted graphs. v1 v2

### Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College

Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1

### Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem

### CSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24

CSL851: Algorithmic Graph Theory Semester I 2013-2014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have

### A Study of Sufficient Conditions for Hamiltonian Cycles

DeLeon 1 A Study of Sufficient Conditions for Hamiltonian Cycles Melissa DeLeon Department of Mathematics and Computer Science Seton Hall University South Orange, New Jersey 07079, U.S.A. ABSTRACT A graph

### 8. Matchings and Factors

8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

### A graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices.

1 Trees A graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices. Theorem 1 For every simple graph G with n 1 vertices, the following

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010

Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows

### D I P L O M A R B E I T. Tree-Decomposition. Graph Minor Theory and Algorithmic Implications. Ausgeführt am Institut für

D I P L O M A R B E I T Tree-Decomposition Graph Minor Theory and Algorithmic Implications Ausgeführt am Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien unter Anleitung

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

### Computational Geometry

Motivation 1 Motivation Polygons and visibility Visibility in polygons Triangulation Proof of the Art gallery theorem Two points in a simple polygon can see each other if their connecting line segment

### THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed

### On the crossing number of K m,n

On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known

### UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a reader-friendly style, it covers the types of graphs, their properties, trees, graph traversability,

### 2.1 You need to know how graphs and networks can be used to create mathematical

fter completing this chapter you should: 1 know how graphs and networks can be used to create mathematical models 2 know some basic terminology used in graph theory 3 know some special types of graph 4

### 6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

### On Pebbling Graphs by their Blocks

On Pebbling Graphs by their Blocks Dawn Curtis, Taylor Hines, Glenn Hurlbert, Tatiana Moyer Department of Mathematics and Statistics Arizona State University, Tempe, AZ 85287-1804 November 19, 2008 dawn.curtis@asu.edu

### Unicyclic Graphs with Given Number of Cut Vertices and the Maximal Merrifield - Simmons Index

Filomat 28:3 (2014), 451 461 DOI 10.2298/FIL1403451H Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Unicyclic Graphs with Given Number

### Extremal Graphs without Three-Cycles or Four-Cycles

Extremal Graphs without Three-Cycles or Four-Cycles David K. Garnick Department of Computer Science Bowdoin College Brunswick, Maine 04011 Y. H. Harris Kwong Department of Mathematics and Computer Science

### 9.1. Vertex colouring. 9. Graph colouring. Vertex colouring.

Vertex colouring. 9. Graph colouring k-vertex-critical graphs Approximation algorithms Upper bounds for the vertex chromatic number Brooks Theorem and Hajós Conjecture Chromatic Polynomials Colouring of

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### TREES IN SET THEORY SPENCER UNGER

TREES IN SET THEORY SPENCER UNGER 1. Introduction Although I was unaware of it while writing the first two talks, my talks in the graduate student seminar have formed a coherent series. This talk can be

### Oriented Diameter of Graphs with Diameter 3

Oriented Diameter of Graphs with Diameter 3 Peter K. Kwok, Qi Liu, Douglas B. West revised November, 2008 Abstract In 1978, Chvátal and Thomassen proved that every 2-edge-connected graph with diameter

### Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

CS-62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of