The vertex set is a finite nonempty set. The edge set may be empty, but otherwise its elements are two-element subsets of the vertex set.
|
|
- Ophelia Lawrence
- 5 years ago
- Views:
Transcription
1 Math 3336 Section 10.2 Graph terminology and Special Types of Graphs Definition: A graph is an object consisting of two sets called its vertex set and its edge set. The vertex set is a finite nonempty set. The edge set may be empty, but otherwise its elements are two-element subsets of the vertex set. Definition: The elements of the vertex set are called vertices and the elements of the edge set are called edges. Definitions: If {uu, vv} is an edge of a graph we say {uu, vv} joins or connects the vertices uu and vv, and that uu and vv are adjacent to each other. The edge {uu, vv} is incident to each uu and vv. Two edges incident to the same vertex are called adjacent edges. A vertex incident to no edges is called isolated. The set of all neighbors of a vertex vv of GG = (VV, EE), denoted by NN(vv), is called the neighborhood of vv. If AA is a subset of VV, we denote by NN(AA) the set of all vertices in GG that are adjacent to at least one vertex in AA. The degree of a vertex in a undirected graph is the number of edges incident with it, except that a loop at a vertex contributes two to the degree of that vertex. The degree of the vertex vv is denoted by dddddd(vv). Example: Page 1 of 7
2 Theorem (Handshaking Theorem): If GG = (VV, EE) is an undirected graph with mm edges, then 2mm = deg (vv) Proof: vv VV. Example: How many edges are there in a graph with 10 vertices of degree six? Solution: Example: If a graph has 5 vertices, can each vertex have degree 3? Solution: Page 2 of 7
3 Theorem: An undirected graph has an even number of vertices of odd degree. Proof: Special Types of Graphs 1. A complete graph on nn vertices, denoted by KK nn, is the graph that have all possible edges. Theorem: The number of edges in a complete graph KK nn is given by the formula vv(vv 1) ee = 2 Proof: Page 3 of 7
4 2. If nn is an integer greater than or equal to 3, the cyclic graph on nn vertices, denoted CC nn, is the graph having vertex set {1, 2, 3,, nn} and edge set {{1,2}, {2, 3}, {3, 4},, {nn 1, nn}}. 3. A wheel WW nn is obtained by adding an additional vertex to a cycle CC nn for nn 3 and connecting this new vertex to each of the nn vertices incc nn by new edges. 4. A simple graph GG is bipartite if VV can be partitioned into two disjoint subsets VV 1 and VV 2 such that every edge connects a vertex in VV 1 and a vertex in VV 2. In other words, there are no edges which connect two vertices in VV 1 or in VV 2. It is not hard to show that an equivalent definition of a bipartite graph is a graph where it is possible to color the vertices red or blue so that no two adjacent vertices are the same color. Example: Determine if G and H are bipartite graphs. Page 4 of 7
5 Example: Show that CC 6 is bipartite. Example: Show that C 3 is not bipartite. 5. A complete bipartite graph KK mm,nn is a graph that has its vertex set partitioned into two subsets VV 1 of size m and VV 2 of size n such that there is an edge from every vertex in VV 1 to every vertex in VV 2. Page 5 of 7
6 Bipartite graphs are used to model applications that involve matching the elements of one set to elements in another, for example: Job assignments - vertices represent the jobs and the employees, edges link employees with those jobs they have been trained to do. A common goal is to match jobs to employees so that the most jobs are done. Marriage - vertices represent the men and the women and edges link a man and a woman if they are an acceptable spouse. We may wish to find the largest number of possible marriages. HALL S MARRAGE THEOREM Suppose that there are m men and n women on an island. Each person has a list of members of the opposite gender acceptable as a spouse. We construct a bipartite graph GG(VV 1, VV 2 ), where VV 1 is the set of men and VV 2 is the set of women. There is an edge between a man and a women if they find each other acceptable as a spouse. A matching in this graph consists of a set of edges, where each pair of endpoints of an edge is a husband- wife pair. A maximum matching is a largest possible set of married couples. A complete matching of VV 1 is a set of married couples where every man is married, but possibly not all women. Page 6 of 7
7 HALL S MARRAGE THEOREM The bipartite graph GG = (VV, EE) with partition (VV 1, VV 2 ) has a complete matching from VV 1 to VV 2 if and only if NN(AA) AA for all subsets AA of VV 1. Page 7 of 7
DEFINITION OF GRAPH GRAPH THEORY GRAPHS ACCORDING TO THEIR VERTICES AND EDGES EXAMPLE GRAPHS ACCORDING TO THEIR VERTICES AND EDGES
DEFINITION OF GRAPH GRAPH THEORY Prepared by Engr. JP Timola Reference: Discrete Math by Kenneth H. Rosen A graph G = (V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each
More informationIntroduction to Graphs
Graphs Introduction to Graphs Graph Terminology Directed Graphs Special Graphs Graph Coloring Representing Graphs Connected Graphs Connected Component Reading (Epp s textbook) 10.1-10.3 1 Introduction
More informationCPCS Discrete Structures 1
Let us switch to a new topic: Graphs CPCS 222 - Discrete Structures 1 Introduction to Graphs Definition: A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs
More informationSection 8.2 Graph Terminology. Undirected Graphs. Definition: Two vertices u, v in V are adjacent or neighbors if there is an edge e between u and v.
Section 8.2 Graph Terminology Undirected Graphs Definition: Two vertices u, v in V are adjacent or neighbors if there is an edge e between u and v. The edge e connects u and v. The vertices u and v are
More informationTheorem 3.1 (Berge) A matching M in G is maximum if and only if there is no M- augmenting path.
3 Matchings Hall s Theorem Matching: A matching in G is a subset M E(G) so that no edge in M is a loop, and no two edges in M are incident with a common vertex. A matching M is maximal if there is no matching
More information(5.2) 151 Math Exercises. Graph Terminology and Special Types of Graphs. Malek Zein AL-Abidin
King Saud University College of Science Department of Mathematics 151 Math Exercises (5.2) Graph Terminology and Special Types of Graphs Malek Zein AL-Abidin ه Basic Terminology First, we give some terminology
More informationMatchings and Covers in bipartite graphs
Matchings and Covers in bipartite graphs A bipartite graph, also called a bigraph, is a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent.
More informationInduction Review. Graphs. EECS 310: Discrete Math Lecture 5 Graph Theory, Matching. Common Graphs. a set of edges or collection of two-elt subsets
EECS 310: Discrete Math Lecture 5 Graph Theory, Matching Reading: MIT OpenCourseWare 6.042 Chapter 5.1-5.2 Induction Review Basic Induction: Want to prove P (n). Prove base case P (1). Prove P (n) P (n+1)
More informationGraphs. Introduction To Graphs: Exercises. Definitions:
Graphs Eng.Jehad Aldahdooh Introduction To Graphs: Definitions: A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated
More informationAssignment 4 Solutions of graph problems
Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of
More information11.4 Bipartite Multigraphs
11.4 Bipartite Multigraphs Introduction Definition A graph G is bipartite if we can partition the vertices into two disjoint subsets U and V such that every edge of G has one incident vertex in U and the
More informationAssignment and Matching
Assignment and Matching By Geetika Rana IE 680 Dept of Industrial Engineering 1 Contents Introduction Bipartite Cardinality Matching Problem Bipartite Weighted Matching Problem Stable Marriage Problem
More informationMath.3336: Discrete Mathematics. Chapter 10 Graph Theory
Math.3336: Discrete Mathematics Chapter 10 Graph Theory Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall
More informationGraphs. Pseudograph: multiple edges and loops allowed
Graphs G = (V, E) V - set of vertices, E - set of edges Undirected graphs Simple graph: V - nonempty set of vertices, E - set of unordered pairs of distinct vertices (no multiple edges or loops) Multigraph:
More informationNumber Theory and Graph Theory
1 Number Theory and Graph Theory Chapter 6 Basic concepts and definitions of graph theory By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com
More informationMATH 682 Notes Combinatorics and Graph Theory II
1 Matchings A popular question to be asked on graphs, if graphs represent some sort of compatability or association, is how to associate as many vertices as possible into well-matched pairs. It is to this
More informationMatching Algorithms. Proof. If a bipartite graph has a perfect matching, then it is easy to see that the right hand side is a necessary condition.
18.433 Combinatorial Optimization Matching Algorithms September 9,14,16 Lecturer: Santosh Vempala Given a graph G = (V, E), a matching M is a set of edges with the property that no two of the edges have
More informationMa/CS 6b Class 5: Graph Connectivity
Ma/CS 6b Class 5: Graph Connectivity By Adam Sheffer Communications Network We are given a set of routers and wish to connect pairs of them to obtain a connected communications network. The network should
More informationCHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism
CHAPTER 2 Graphs 1. Introduction to Graphs and Graph Isomorphism 1.1. The Graph Menagerie. Definition 1.1.1. A simple graph G = (V, E) consists of a set V of vertices and a set E of edges, represented
More informationGraph Theory II. Po-Shen Loh. June edges each. Solution: Spread the n vertices around a circle. Take parallel classes.
Graph Theory II Po-Shen Loh June 009 1 Warm-up 1. Let n be odd. Partition the edge set of K n into n matchings with n 1 edges each. Solution: Spread the n vertices around a circle. Take parallel classes..
More informationMath 170- Graph Theory Notes
1 Math 170- Graph Theory Notes Michael Levet December 3, 2018 Notation: Let n be a positive integer. Denote [n] to be the set {1, 2,..., n}. So for example, [3] = {1, 2, 3}. To quote Bud Brown, Graph theory
More informationMatching and Covering
Matching and Covering Matchings A matching of size k in a graph G is a set of k pairwise disjoint edges The vertices belonging to the edges of a matching are saturated by the matching; the others are unsaturated
More informationVarying Applications (examples)
Graph Theory Varying Applications (examples) Computer networks Distinguish between two chemical compounds with the same molecular formula but different structures Solve shortest path problems between cities
More information5 Graphs
5 Graphs jacques@ucsd.edu Some of the putnam problems are to do with graphs. They do not assume more than a basic familiarity with the definitions and terminology of graph theory. 5.1 Basic definitions
More informationMC302 GRAPH THEORY Thursday, 10/24/13
MC302 GRAPH THEORY Thursday, 10/24/13 Today: Return, discuss HW 3 From last time: Greedy Algorithms for TSP Matchings and Augmenting Paths HW 4 will be posted by tomorrow Reading: [CH] 4.1 Exercises: [CH]
More informationFundamentals of Graph Theory MATH Fundamentals of Graph Theory. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Fundamentals of Graph Theory Benjamin V.C. Collins James A. Swenson The seven bridges of Königsberg Map: Merian-Erben [Public domain], via Wikimedia Commons The seven bridges of Königsberg Map:
More information5 Matchings in Bipartite Graphs and Their Applications
5 Matchings in Bipartite Graphs and Their Applications 5.1 Matchings Definition 5.1 A matching M in a graph G is a set of edges of G, none of which is a loop, such that no two edges in M have a common
More informationCS 311 Discrete Math for Computer Science Dr. William C. Bulko. Graphs
CS 311 Discrete Math for Computer Science Dr. William C. Bulko Graphs 2014 Definitions Definition: A graph G = (V,E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge
More informationMatching. Algorithms and Networks
Matching Algorithms and Networks This lecture Matching: problem statement and applications Bipartite matching (recap) Matching in arbitrary undirected graphs: Edmonds algorithm Diversion: generalized tic-tac-toe
More informationMath 575 Exam 3. (t). What is the chromatic number of G?
Math 575 Exam 3 Name 1 (a) Draw the Grötsch graph See your notes (b) Suppose that G is a graph having 6 vertices and 9 edges and that the chromatic polynomial of G is given below Fill in the missing coefficients
More informationDefinition For vertices u, v V (G), the distance from u to v, denoted d(u, v), in G is the length of a shortest u, v-path. 1
Graph fundamentals Bipartite graph characterization Lemma. If a graph contains an odd closed walk, then it contains an odd cycle. Proof strategy: Consider a shortest closed odd walk W. If W is not a cycle,
More informationGraph Theory: Matchings and Factors
Graph Theory: Matchings and Factors Pallab Dasgupta, Professor, Dept. of Computer Sc. and Engineering, IIT Kharagpur pallab@cse.iitkgp.ernet.in Matchings A matching of size k in a graph G is a set of k
More informationDiscrete Mathematics and Probability Theory Summer 2016 Dinh, Psomas, and Ye HW 2
CS 70 Discrete Mathematics and Probability Theory Summer 2016 Dinh, Psomas, and Ye HW 2 Due Tuesday July 5 at 1:59PM 1. (8 points: 3/5) Hit or miss For each of the claims and proofs below, state whether
More informationON A WEAKER VERSION OF SUM LABELING OF GRAPHS
ON A WEAKER VERSION OF SUM LABELING OF GRAPHS IMRAN JAVAID, FARIHA KHALID, ALI AHMAD and M. IMRAN Communicated by the former editorial board In this paper, we introduce super weak sum labeling and weak
More informationIntroduction to Graph Theory
Introduction to Graph Theory Tandy Warnow January 20, 2017 Graphs Tandy Warnow Graphs A graph G = (V, E) is an object that contains a vertex set V and an edge set E. We also write V (G) to denote the vertex
More informationCMSC 380. Graph Terminology and Representation
CMSC 380 Graph Terminology and Representation GRAPH BASICS 2 Basic Graph Definitions n A graph G = (V,E) consists of a finite set of vertices, V, and a finite set of edges, E. n Each edge is a pair (v,w)
More informationAMS /672: Graph Theory Homework Problems - Week V. Problems to be handed in on Wednesday, March 2: 6, 8, 9, 11, 12.
AMS 550.47/67: Graph Theory Homework Problems - Week V Problems to be handed in on Wednesday, March : 6, 8, 9,,.. Assignment Problem. Suppose we have a set {J, J,..., J r } of r jobs to be filled by a
More informationREU 2006 Discrete Math Lecture 5
REU 2006 Discrete Math Lecture 5 Instructor: László Babai Scribe: Megan Guichard Editors: Duru Türkoğlu and Megan Guichard June 30, 2006. Last updated July 3, 2006 at 11:30pm. 1 Review Recall the definitions
More informationFoundations of Discrete Mathematics
Foundations of Discrete Mathematics Chapters 9 By Dr. Dalia M. Gil, Ph.D. Graphs Graphs are discrete structures consisting of vertices and edges that connect these vertices. Graphs A graph is a pair (V,
More informationMath 778S Spectral Graph Theory Handout #2: Basic graph theory
Math 778S Spectral Graph Theory Handout #: Basic graph theory Graph theory was founded by the great Swiss mathematician Leonhard Euler (1707-178) after he solved the Königsberg Bridge problem: Is it possible
More informationMath 776 Graph Theory Lecture Note 1 Basic concepts
Math 776 Graph Theory Lecture Note 1 Basic concepts Lectured by Lincoln Lu Transcribed by Lincoln Lu Graph theory was founded by the great Swiss mathematician Leonhard Euler (1707-178) after he solved
More informationProposition 1. The edges of an even graph can be split (partitioned) into cycles, no two of which have an edge in common.
Math 3116 Dr. Franz Rothe June 5, 2012 08SUM\3116_2012t1.tex Name: Use the back pages for extra space 1 Solution of Test 1.1 Eulerian graphs Proposition 1. The edges of an even graph can be split (partitioned)
More informationMatchings, Ramsey Theory, And Other Graph Fun
Matchings, Ramsey Theory, And Other Graph Fun Evelyne Smith-Roberge University of Waterloo April 5th, 2017 Recap... In the last two weeks, we ve covered: What is a graph? Eulerian circuits Hamiltonian
More informationAdjacent: Two distinct vertices u, v are adjacent if there is an edge with ends u, v. In this case we let uv denote such an edge.
1 Graph Basics What is a graph? Graph: a graph G consists of a set of vertices, denoted V (G), a set of edges, denoted E(G), and a relation called incidence so that each edge is incident with either one
More informationPACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS
PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS PAUL BALISTER Abstract It has been shown [Balister, 2001] that if n is odd and m 1,, m t are integers with m i 3 and t i=1 m i = E(K n) then K n can be decomposed
More informationIntroduction III. Graphs. Motivations I. Introduction IV
Introduction I Graphs Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Graph theory was introduced in the 18th century by Leonhard Euler via the Königsberg
More information1 Matchings in Graphs
Matchings in Graphs J J 2 J 3 J 4 J 5 J J J 6 8 7 C C 2 C 3 C 4 C 5 C C 7 C 8 6 J J 2 J 3 J 4 J 5 J J J 6 8 7 C C 2 C 3 C 4 C 5 C C 7 C 8 6 Definition Two edges are called independent if they are not adjacent
More informationCPS 102: Discrete Mathematics. Quiz 3 Date: Wednesday November 30, Instructor: Bruce Maggs NAME: Prob # Score. Total 60
CPS 102: Discrete Mathematics Instructor: Bruce Maggs Quiz 3 Date: Wednesday November 30, 2011 NAME: Prob # Score Max Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1 Problem 1 [10 points] Find a minimum-cost
More informationCMSC Honors Discrete Mathematics
CMSC 27130 Honors Discrete Mathematics Lectures by Alexander Razborov Notes by Justin Lubin The University of Chicago, Autumn 2017 1 Contents I Number Theory 4 1 The Euclidean Algorithm 4 2 Mathematical
More informationDomination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes
Domination, Independence and Other Numbers Associated With the Intersection Graph of a Set of Half-planes Leonor Aquino-Ruivivar Mathematics Department, De La Salle University Leonorruivivar@dlsueduph
More informationResearch Question Presentation on the Edge Clique Covers of a Complete Multipartite Graph. Nechama Florans. Mentor: Dr. Boram Park
Research Question Presentation on the Edge Clique Covers of a Complete Multipartite Graph Nechama Florans Mentor: Dr. Boram Park G: V 5 Vertex Clique Covers and Edge Clique Covers: Suppose we have a graph
More informationA graph is finite if its vertex set and edge set are finite. We call a graph with just one vertex trivial and all other graphs nontrivial.
2301-670 Graph theory 1.1 What is a graph? 1 st semester 2550 1 1.1. What is a graph? 1.1.2. Definition. A graph G is a triple (V(G), E(G), ψ G ) consisting of V(G) of vertices, a set E(G), disjoint from
More informationCopyright 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin Introduction to the Design & Analysis of Algorithms, 2 nd ed., Ch.
Iterative Improvement Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible
More informationCS473-Algorithms I. Lecture 13-A. Graphs. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorithms I Lecture 3-A Graphs Graphs A directed graph (or digraph) G is a pair (V, E), where V is a finite set, and E is a binary relation on V The set V: Vertex set of G The set E: Edge set of
More informationLet G = (V, E) be a graph. If u, v V, then u is adjacent to v if {u, v} E. We also use the notation u v to denote that u is adjacent to v.
Graph Adjacent Endpoint of an edge Incident Neighbors of a vertex Degree of a vertex Theorem Graph relation Order of a graph Size of a graph Maximum and minimum degree Let G = (V, E) be a graph. If u,
More informationMath236 Discrete Maths with Applications
Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 19 Degree Sequences Let G be a graph with vertex set V (G) = {v 1, v 2, v
More informationArtificial Intelligence
Artificial Intelligence Graph theory G. Guérard Department of Nouvelles Energies Ecole Supérieur d Ingénieurs Léonard de Vinci Lecture 1 GG A.I. 1/37 Outline 1 Graph theory Undirected and directed graphs
More informationBasics of Graph Theory
Basics of Graph Theory 1 Basic notions A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Simple graphs have their
More informationModule 7. Independent sets, coverings. and matchings. Contents
Module 7 Independent sets, coverings Contents and matchings 7.1 Introduction.......................... 152 7.2 Independent sets and coverings: basic equations..... 152 7.3 Matchings in bipartite graphs................
More informationJessica Su (some parts copied from CLRS / last quarter s notes)
1 Max flow Consider a directed graph G with positive edge weights c that define the capacity of each edge. We can identify two special nodes in G: the source node s and the sink node t. We want to find
More informationConnecting Statements. Today. First there was logic jumping forward.. ..and then proofs and then induction...
Today Review for Midterm. First there was logic... A statement is a true or false. Statements? 3 = 4 1? Statement! 3 = 5? Statement! 3? Not a statement! n = 3? Not a statement...but a predicate. Predicate:
More informationOn vertex-coloring edge-weighting of graphs
Front. Math. China DOI 10.1007/s11464-009-0014-8 On vertex-coloring edge-weighting of graphs Hongliang LU 1, Xu YANG 1, Qinglin YU 1,2 1 Center for Combinatorics, Key Laboratory of Pure Mathematics and
More informationDO NOT RE-DISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 104, Graph Theory Homework 2 Solutions February 7, 2013 Introduction to Graph Theory, West Section 1.2: 26, 38, 42 Section 1.3: 14, 18 Section 2.1: 26, 29, 30 DO NOT RE-DISTRIBUTE
More informationGraph Theory. Connectivity, Coloring, Matching. Arjun Suresh 1. 1 GATE Overflow
Graph Theory Connectivity, Coloring, Matching Arjun Suresh 1 1 GATE Overflow GO Classroom, August 2018 Thanks to Subarna/Sukanya Das for wonderful figures Arjun, Suresh (GO) Graph Theory GATE 2019 1 /
More informationGraph theory. Po-Shen Loh. June We begin by collecting some basic facts which can be proved via bare-hands techniques.
Graph theory Po-Shen Loh June 013 1 Basic results We begin by collecting some basic facts which can be proved via bare-hands techniques. 1. The sum of all of the degrees is equal to twice the number of
More informationAssignment 1 Introduction to Graph Theory CO342
Assignment 1 Introduction to Graph Theory CO342 This assignment will be marked out of a total of thirty points, and is due on Thursday 18th May at 10am in class. Throughout the assignment, the graphs are
More informationSome Elementary Lower Bounds on the Matching Number of Bipartite Graphs
Some Elementary Lower Bounds on the Matching Number of Bipartite Graphs Ermelinda DeLaViña and Iride Gramajo Department of Computer and Mathematical Sciences University of Houston-Downtown Houston, Texas
More informationGraceful Labeling for Cycle of Graphs
International Journal of Mathematics Research. ISSN 0976-5840 Volume 6, Number (014), pp. 173 178 International Research Publication House http://www.irphouse.com Graceful Labeling for Cycle of Graphs
More informationBalls Into Bins Model Occupancy Problems
Balls Into Bins Model Occupancy Problems Assume that m balls are thrown randomly is n bins, i.e., the location of each ball is independently and uniformly chosen at random from the n possibilities. How
More informationOn Strongly *-Graphs
Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences 54 (2): 179 195 (2017) Copyright Pakistan Academy of Sciences ISSN: 2518-4245 (print), 2518-4253 (online) Pakistan
More informationGraph Theory CS/Math231 Discrete Mathematics Spring2015
1 Graphs Definition 1 A directed graph (or digraph) G is a pair (V, E), where V is a finite set and E is a binary relation on V. The set V is called the vertex set of G, and its elements are called vertices
More informationBinding Number of Some Special Classes of Trees
International J.Math. Combin. Vol.(206), 76-8 Binding Number of Some Special Classes of Trees B.Chaluvaraju, H.S.Boregowda 2 and S.Kumbinarsaiah 3 Department of Mathematics, Bangalore University, Janana
More informationThe Geodesic Integral on Medial Graphs
The Geodesic Integral on Medial Graphs Kolya Malkin August 013 We define the geodesic integral defined on paths in the duals of medial graphs on surfaces and use it to study lens elimination and connection
More informationLecture 6: Graph Properties
Lecture 6: Graph Properties Rajat Mittal IIT Kanpur In this section, we will look at some of the combinatorial properties of graphs. Initially we will discuss independent sets. The bulk of the content
More informationON THE STRUCTURE OF SELF-COMPLEMENTARY GRAPHS ROBERT MOLINA DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE ALMA COLLEGE ABSTRACT
ON THE STRUCTURE OF SELF-COMPLEMENTARY GRAPHS ROBERT MOLINA DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE ALMA COLLEGE ABSTRACT A graph G is self complementary if it is isomorphic to its complement G.
More information9.5 Equivalence Relations
9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same
More informationBinary Relations McGraw-Hill Education
Binary Relations A binary relation R from a set A to a set B is a subset of A X B Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B. We can also represent
More informationEDGE MAXIMAL GRAPHS CONTAINING NO SPECIFIC WHEELS. Jordan Journal of Mathematics and Statistics (JJMS) 8(2), 2015, pp I.
EDGE MAXIMAL GRAPHS CONTAINING NO SPECIFIC WHEELS M.S.A. BATAINEH (1), M.M.M. JARADAT (2) AND A.M.M. JARADAT (3) A. Let k 4 be a positive integer. Let G(n; W k ) denote the class of graphs on n vertices
More informationCharacterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4)
S-72.2420/T-79.5203 Basic Concepts 1 S-72.2420/T-79.5203 Basic Concepts 3 Characterizing Graphs (1) Characterizing Graphs (3) Characterizing a class G by a condition P means proving the equivalence G G
More informationDiscrete mathematics , Fall Instructor: prof. János Pach
Discrete mathematics 2016-2017, Fall Instructor: prof. János Pach - covered material - Lecture 1. Counting problems To read: [Lov]: 1.2. Sets, 1.3. Number of subsets, 1.5. Sequences, 1.6. Permutations,
More information1 The Arthur-Merlin Story
Comp 260: Advanced Algorithms Tufts University, Spring 2011 Prof. Lenore Cowen Scribe: Andrew Winslow Lecture 1: Perfect and Stable Marriages 1 The Arthur-Merlin Story In the land ruled by the legendary
More informationGraph Theory S 1 I 2 I 1 S 2 I 1 I 2
Graph Theory S I I S S I I S Graphs Definition A graph G is a pair consisting of a vertex set V (G), and an edge set E(G) ( ) V (G). x and y are the endpoints of edge e = {x, y}. They are called adjacent
More informationKey Graph Theory Theorems
Key Graph Theory Theorems Rajesh Kumar MATH 239 Intro to Combinatorics August 19, 2008 3.3 Binary Trees 3.3.1 Problem (p.82) Determine the number, t n, of binary trees with n edges. The number of binary
More informationPartitioning Complete Multipartite Graphs by Monochromatic Trees
Partitioning Complete Multipartite Graphs by Monochromatic Trees Atsushi Kaneko, M.Kano 1 and Kazuhiro Suzuki 1 1 Department of Computer and Information Sciences Ibaraki University, Hitachi 316-8511 Japan
More informationZhibin Huang 07. Juni Zufällige Graphen
Zhibin Huang 07. Juni 2010 Seite 2 Contents The Basic Method The Probabilistic Method The Ramsey Number R( k, l) Linearity of Expectation Basics Splitting Graphs The Probabilistic Lens: High Girth and
More informationGRAPH THEORY and APPLICATIONS. Factorization Domination Indepence Clique
GRAPH THEORY and APPLICATIONS Factorization Domination Indepence Clique Factorization Factor A factor of a graph G is a spanning subgraph of G, not necessarily connected. G is the sum of factors G i, if:
More informationBipartite graphs unique perfect matching.
Generation of graphs Bipartite graphs unique perfect matching. In this section, we assume G = (V, E) bipartite connected graph. The following theorem states that if G has unique perfect matching, then
More informationConstraint Programming. Global Constraints. Amira Zaki Prof. Dr. Thom Frühwirth. University of Ulm WS 2012/2013
Global Constraints Amira Zaki Prof. Dr. Thom Frühwirth University of Ulm WS 2012/2013 Amira Zaki & Thom Frühwirth University of Ulm Page 1 WS 2012/2013 Overview Classes of Constraints Global Constraints
More informationLecture 7: Bipartite Matching
Lecture 7: Bipartite Matching Bipartite matching Non-bipartite matching What is a Bipartite Matching? Let G=(N,A) be an unrestricted bipartite graph. A subset X of A is said to be a matching if no two
More informationMa/CS 6b Class 2: Matchings
Ma/CS 6b Class 2: Matchings By Adam Sheffer Send anonymous suggestions and complaints from here. Email: adamcandobetter@gmail.com Password: anonymous2 There aren t enough crocodiles in the presentations
More informationMa/CS 6b Class 10: Ramsey Theory
Ma/CS 6b Class 10: Ramsey Theory By Adam Sheffer The Pigeonhole Principle The pigeonhole principle. If n items are put into m containers, such that n > m, then at least one container contains more than
More informationOrdinary Differential Equation (ODE)
Ordinary Differential Equation (ODE) INTRODUCTION: Ordinary Differential Equations play an important role in different branches of science and technology In the practical field of application problems
More informationChapter 2 Graphs. 2.1 Definition of Graphs
Chapter 2 Graphs Abstract Graphs are discrete structures that consist of vertices and edges connecting some of these vertices. Graphs have many applications in Mathematics, Computer Science, Engineering,
More informationCS 441 Discrete Mathematics for CS Lecture 26. Graphs. CS 441 Discrete mathematics for CS. Final exam
CS 441 Discrete Mathematics for CS Lecture 26 Graphs Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Final exam Saturday, April 26, 2014 at 10:00-11:50am The same classroom as lectures The exam
More informationChapter 6 GRAPH COLORING
Chapter 6 GRAPH COLORING A k-coloring of (the vertex set of) a graph G is a function c : V (G) {1, 2,..., k} such that c (u) 6= c (v) whenever u is adjacent to v. Ifak-coloring of G exists, then G is called
More informationNEIGHBOURHOOD SUM CORDIAL LABELING OF GRAPHS
NEIGHBOURHOOD SUM CORDIAL LABELING OF GRAPHS A. Muthaiyan # and G. Bhuvaneswari * Department of Mathematics, Government Arts and Science College, Veppanthattai, Perambalur - 66, Tamil Nadu, India. P.G.
More informationWUCT121. Discrete Mathematics. Graphs
WUCT121 Discrete Mathematics Graphs WUCT121 Graphs 1 Section 1. Graphs 1.1. Introduction Graphs are used in many fields that require analysis of routes between locations. These areas include communications,
More informationNetwork flows and Menger s theorem
Network flows and Menger s theorem Recall... Theorem (max flow, min cut strong duality). Let G be a network. The maximum value of a flow equals the minimum capacity of a cut. We prove this strong duality
More informationSources for this lecture. 3. Matching in bipartite and general graphs. Symmetric difference
S-72.2420 / T-79.5203 Matching in bipartite and general graphs 1 3. Matching in bipartite and general graphs Let G be a graph. A matching M in G is a set of nonloop edges with no shared endpoints. Let
More informationCrossing bridges. Crossing bridges Great Ideas in Theoretical Computer Science. Lecture 12: Graphs I: The Basics. Königsberg (Prussia)
15-251 Great Ideas in Theoretical Computer Science Lecture 12: Graphs I: The Basics February 22nd, 2018 Crossing bridges Königsberg (Prussia) Now Kaliningrad (Russia) Is there a way to walk through the
More information