MAS341: Graph Theory
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Chapter 7 Planar Graphs
Chapter 7 Planar graphs In full: 7.1–7.3 Parts of: 7.4, 7.6–7.8 Skip: 7.5 Prof. Tesler Math 154 Winter 2020 Prof. Tesler Ch. 7: Planar Graphs Math 154 / Winter 2020 1 / 52 Planar graphs Definition A planar embedding of a graph is a drawing of the graph in the plane without edges crossing. A graph is planar if a planar embedding of it exists. Consider two drawings of the graph K4: V = f1, 2, 3, 4g E = f1, 2g , f1, 3g , f1, 4g , f2, 3g , f2, 4g , f3, 4g 1 2 1 2 3 4 3 4 Non−planar embedding Planar embedding The abstract graph K4 is planar because it can be drawn in the plane without crossing edges. Prof. Tesler Ch. 7: Planar Graphs Math 154 / Winter 2020 2 / 52 How about K5? Both of these drawings of K5 have crossing edges. We will develop methods to prove that K5 is not a planar graph, and to characterize what graphs are planar. Prof. Tesler Ch. 7: Planar Graphs Math 154 / Winter 2020 3 / 52 Euler’s Theorem on Planar Graphs Let G be a connected planar graph (drawn w/o crossing edges). Define V = number of vertices E = number of edges F = number of faces, including the “infinite” face Then V - E + F = 2. Note: This notation conflicts with standard graph theory notation V and E for the sets of vertices and edges. Alternately, use jV(G)j - jE(G)j + jF(G)j = 2. Example face 3 V = 4 E = 6 face 1 F = 4 face 4 (infinite face) face 2 V - E + F = 4 - 6 + 4 = 2 Prof. -
COMP 761: Lecture 9 – Graph Theory I
COMP 761: Lecture 9 – Graph Theory I David Rolnick September 23, 2020 David Rolnick COMP 761: Graph Theory I Sep 23, 2020 1 / 27 Problem In Königsberg (now called Kaliningrad) there are some bridges (shown below). Is it possible to cross over all of them in some order, without crossing any bridge more than once? (Please don’t post your ideas in the chat just yet, we’ll discuss the problem soon in class.) David Rolnick COMP 761: Graph Theory I Sep 23, 2020 2 / 27 Problem Set 2 will be due on Oct. 9 Vincent’s optional list of practice problems available soon on Slack (we can give feedback on your solutions, but they will not count towards a grade) Course Announcements David Rolnick COMP 761: Graph Theory I Sep 23, 2020 3 / 27 Vincent’s optional list of practice problems available soon on Slack (we can give feedback on your solutions, but they will not count towards a grade) Course Announcements Problem Set 2 will be due on Oct. 9 David Rolnick COMP 761: Graph Theory I Sep 23, 2020 3 / 27 Course Announcements Problem Set 2 will be due on Oct. 9 Vincent’s optional list of practice problems available soon on Slack (we can give feedback on your solutions, but they will not count towards a grade) David Rolnick COMP 761: Graph Theory I Sep 23, 2020 3 / 27 A graph G is defined by a set V of vertices and a set E of edges, where the edges are (unordered) pairs of vertices fv; wg. -
An Introduction to Graph Colouring
An Introduction to Graph Colouring Evelyne Smith-Roberge University of Waterloo March 29, 2017 Recap... Last week, we covered: I What is a graph? I Eulerian circuits I Hamiltonian Cycles I Planarity Reminder: A graph G is: I a set V (G) of objects called vertices together with: I a set E(G), of what we call called edges. An edge is an unordered pair of vertices. We call two vertices adjacent if they are connected by an edge. Today, we'll get into... I Planarity in more detail I The four colour theorem I Vertex Colouring I Edge Colouring Recall... Planarity We said last week that a graph is planar if it can be drawn in such a way that no edges cross. These areas, including the infinite area surrounding the graph, are called faces. We denote the set of faces of a graph G by F (G). Planarity You'll notice the edges of planar graphs cut up our space into different sections. ) Planarity You'll notice the edges of planar graphs cut up our space into different sections. ) These areas, including the infinite area surrounding the graph, are called faces. We denote the set of faces of a graph G by F (G). Degree of a Vertex Last week, we defined the degree of a vertex to be the number of edges that had that vertex as an endpoint. In this graph, for example, each of the vertices has degree 3. Degree of a Face Similarly, we can define the degree of a face. The degree of a face is the number of edges that make up the boundary of that face. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
Clay Mathematics Institute 2005 James A
Contents Clay Mathematics Institute 2005 James A. Carlson Letter from the President 2 The Prize Problems The Millennium Prize Problems 3 Recognizing Achievement 2005 Clay Research Awards 4 CMI Researchers Summary of 2005 6 Workshops & Conferences CMI Programs & Research Activities Student Programs Collected Works James Arthur Archive 9 Raoul Bott Library CMI Profile Interview with Research Fellow 10 Maria Chudnovsky Feature Article Can Biology Lead to New Theorems? 13 by Bernd Sturmfels CMI Summer Schools Summary 14 Ricci Flow, 3–Manifolds, and Geometry 15 at MSRI Program Overview CMI Senior Scholars Program 17 Institute News Euclid and His Heritage Meeting 18 Appointments & Honors 20 CMI Publications Selected Articles by Research Fellows 27 Books & Videos 28 About CMI Profile of Bow Street Staff 30 CMI Activities 2006 Institute Calendar 32 2005 Euclid: www.claymath.org/euclid James Arthur Collected Works: www.claymath.org/cw/arthur Hanoi Institute of Mathematics: www.math.ac.vn Ramanujan Society: www.ramanujanmathsociety.org $.* $MBZ.BUIFNBUJDT*OTUJUVUF ".4 "NFSJDBO.BUIFNBUJDBM4PDJFUZ In addition to major,0O"VHVTU BUUIFTFDPOE*OUFSOBUJPOBM$POHSFTTPG.BUIFNBUJDJBOT ongoing activities such as JO1BSJT %BWJE)JMCFSUEFMJWFSFEIJTGBNPVTMFDUVSFJOXIJDIIFEFTDSJCFE the summer schools,UXFOUZUISFFQSPCMFNTUIBUXFSFUPQMBZBOJOnVFOUJBMSPMFJONBUIFNBUJDBM the Institute undertakes a 5IF.JMMFOOJVN1SJ[F1SPCMFNT SFTFBSDI"DFOUVSZMBUFS PO.BZ BUBNFFUJOHBUUIF$PMMÒHFEF number of smaller'SBODF UIF$MBZ.BUIFNBUJDT*OTUJUVUF $.* BOOPVODFEUIFDSFBUJPOPGB special projects -
HAMILTONICITY in CAYLEY GRAPHS and DIGRAPHS of FINITE ABELIAN GROUPS. Contents 1. Introduction. 1 2. Graph Theory: Introductory
HAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS. MARY STELOW Abstract. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown. Several results are then pre- sented: firstly, it is shown that if G is an abelian group, then G has a Cayley digraph with a Hamiltonian cycle. It is then proven that every Cayley di- graph of a Dedekind group has a Hamiltonian path. Finally, we show that all Cayley graphs of Abelian groups have Hamiltonian cycles. Further results, applications, and significance of the study of Hamiltonicity of Cayley graphs and digraphs are then discussed. Contents 1. Introduction. 1 2. Graph Theory: Introductory Definitions. 2 3. Cayley Graphs and Digraphs. 2 4. Hamiltonian Cycles in Cayley Digraphs of Finite Abelian Groups 5 5. Hamiltonian Paths in Cayley Digraphs of Dedekind Groups. 7 6. Cayley Graphs of Finite Abelian Groups are Guaranteed a Hamiltonian Cycle. 8 7. Conclusion; Further Applications and Research. 9 8. Acknowledgements. 9 References 10 1. Introduction. The topic of Cayley digraphs and graphs exhibits an interesting and important intersection between the world of groups, group theory, and abstract algebra and the world of graph theory and combinatorics. In this paper, I aim to highlight this intersection and to introduce an area in the field for which many basic problems re- main open.The theorems presented are taken from various discrete math journals. Here, these theorems are analyzed and given lengthier treatment in order to be more accessible to those without much background in group or graph theory. -
V=Ih0cpr745fm
MITOCW | watch?v=Ih0cPR745fM The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we have a lecturer, guest lecture two of two, Costis Daskalakis. COSTIS Glad to be back. So let's continue on the path we followed last time. Let me remind DASKALAKIS: you what we did last time, first of all. So I talked about interesting theorems in topology-- Nash, Sperner, and Brouwer. And I defined the corresponding-- so these were theorems in topology. Define the corresponding problems. And because of these existence theorems, the corresponding search problems were total. And then I looked into the problems in NP that are total, and I tried to identify what in these problems make them total and tried to identify combinatorial argument that guarantees the existence of solutions in these problems. Motivated by the argument, which turned out to be a parity argument on directed graphs, I defined the class PPAD, and I introduced the problem of ArithmCircuitSAT, which is PPAD complete, and from which I promised to show a bunch of PPAD hardness deductions this time. So let me remind you the salient points from this list before I keep going. So first of all, the PPAD class has a combinatorial flavor. In the definition of the class, what I'm doing is I'm defining a graph on all possible n-bit strings, so an exponentially large set by providing two circuits, P and N. -
What Is a Polyhedron?
3. POLYHEDRA, GRAPHS AND SURFACES 3.1. From Polyhedra to Graphs What is a Polyhedron? Now that we’ve covered lots of geometry in two dimensions, let’s make things just a little more difficult. We’re going to consider geometric objects in three dimensions which can be made from two-dimensional pieces. For example, you can take six squares all the same size and glue them together to produce the shape which we call a cube. More generally, if you take a bunch of polygons and glue them together so that no side gets left unglued, then the resulting object is usually called a polyhedron.1 The corners of the polygons are called vertices, the sides of the polygons are called edges and the polygons themselves are called faces. So, for example, the cube has 8 vertices, 12 edges and 6 faces. Different people seem to define polyhedra in very slightly different ways. For our purposes, we will need to add one little extra condition — that the volume bound by a polyhedron “has no holes”. For example, consider the shape obtained by drilling a square hole straight through the centre of a cube. Even though the surface of such a shape can be constructed by gluing together polygons, we don’t consider this shape to be a polyhedron, because of the hole. We say that a polyhedron is convex if, for each plane which lies along a face, the polyhedron lies on one side of that plane. So, for example, the cube is a convex polyhedron while the more complicated spec- imen of a polyhedron pictured on the right is certainly not convex. -
The Power of Shaking Hands
The power of shaking hands Dion Gijswijt Although very simple to prove, the handshaking lemma can be a powerful Observe that the number of neighbours of a Hamiltonian path P between s and u is equal to d(u) − 1. It follows that such a Hamiltonian path has odd tool in the hands of a combinatorialist. Here, I will show you some degree in H if and only if u = t. By the handshaking lemma, H has an even number of odd degree nodes, which means that G has an even number colourful applications and pose some challenges to you, dear reader. of Hamiltonian paths between s and t. Let me start by posing a simple question. Perhaps you already know the answer. As an example, consider the graph in Figure 1. Nodes s and t have degree 2 If not, take a minute or two to see if you can solve it! and all other nodes have degree 3. One Hamiltonian path between s and t is indicated. By Smith’s theorem, there are an even number of such paths, hence There are seven people at a party. Is it possible that each of them there must be at least one other such path. Can you find it? shakes hands with exactly three others? In the language of graph theory, we are asking for a graph1 with 7 nodes in which every node has degree 3. The following simple observation will be a central idea in this article. s t Lemma 1 (handshaking lemma). Let G be a graph. -
The Monadic Second-Order Logic of Graphs Xvi: Canonical Graph Decompositions
Logical Methods in Computer Science Vol. 2 (2:2) 2006, pp. 1–46 Submitted Jun. 24, 2005 www.lmcs-online.org Published Mar. 23, 2006 THE MONADIC SECOND-ORDER LOGIC OF GRAPHS XVI: CANONICAL GRAPH DECOMPOSITIONS BRUNO COURCELLE LaBRI, Bordeaux 1 University, 33405 Talence, France e-mail address: [email protected] Abstract. This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decom- position and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs having the same cycle matroid as a given 2-connected graph can be defined from this graph by Monadic Second-Order formulas. 1. Introduction Hierarchical graph decompositions are useful for the construction of efficient algorithms, and also because they give structural descriptions of the considered graphs. Cunningham and Edmonds have proposed in [18] a general framework for defining decompositions of graphs, hypergraphs and matroids. This framework covers many types of decompositions. Of particular interest is the split decomposition of directed and undirected graphs defined by Cunningham in [17]. A hierarchical decomposition of a certain type is canonical if, up to technical details like vertex labellings, there is a unique decomposition of a given graph (or hypergraph, or matroid) of this type. To take well-known examples concerning graphs, the modular decom- position is canonical, whereas, except in particular cases, there is no useful canonical notion of tree-decomposition of minimal tree-width. -
Tree-Decomposition Graph Minor Theory and Algorithmic Implications
DIPLOMARBEIT Tree-Decomposition Graph Minor Theory and Algorithmic Implications Ausgeführt am Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien unter Anleitung von Univ.Prof. Dipl.-Ing. Dr.techn. Michael Drmota durch Miriam Heinz, B.Sc. Matrikelnummer: 0625661 Baumgartenstraße 53 1140 Wien Datum Unterschrift Preface The focus of this thesis is the concept of tree-decomposition. A tree-decomposition of a graph G is a representation of G in a tree-like structure. From this structure it is possible to deduce certain connectivity properties of G. Such information can be used to construct efficient algorithms to solve problems on G. Sometimes problems which are NP-hard in general are solvable in polynomial or even linear time when restricted to trees. Employing the tree-like structure of tree-decompositions these algorithms for trees can be adapted to graphs of bounded tree-width. This results in many important algorithmic applications of tree-decomposition. The concept of tree-decomposition also proves to be useful in the study of fundamental questions in graph theory. It was used extensively by Robertson and Seymour in their seminal work on Wagner’s conjecture. Their Graph Minors series of papers spans more than 500 pages and results in a proof of the graph minor theorem, settling Wagner’s conjecture in 2004. However, it is not only the proof of this deep and powerful theorem which merits mention. Also the concepts and tools developed for the proof have had a major impact on the field of graph theory. Tree-decomposition is one of these spin-offs. Therefore, we will study both its use in the context of graph minor theory and its several algorithmic implications.