MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 10, Monday, September 22.

Slides:

Advertisements

Similar presentations

Trees Chapter 11.

Advertisements

Chapter 8 Topics in Graph Theory

Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.

Edge-Coloring of Graphs On the left we see a 1- factorization of  5, the five-sided prism. Each factor is respresented by its own color. No edges of the.

Chapter 9 Connectivity 连通度. 9.1 Connectivity Consider the following graphs:  G 1 : Deleting any edge makes it disconnected.  G 2 : Cannot be disconnected.

CMPS 2433 Discrete Structures Chapter 5 - Trees R. HALVERSON – MIDWESTERN STATE UNIVERSITY.

Applied Combinatorics, 4th Ed. Alan Tucker

Last time: terminology reminder w Simple graph Vertex = node Edge Degree Weight Neighbours Complete Dual Bipartite Planar Cycle Tree Path Circuit Components.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 19, Friday, October 17.

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

Lists A list is a finite, ordered sequence of data items. Two Implementations –Arrays –Linked Lists.

Graphs and Trees This handout: Trees Minimum Spanning Tree Problem.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 3, Friday, September 5.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 14, Wednesday, October 1.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 15, Friday, October 3.

Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.

Chapter 1 Review Homework (MATH 310#3M):

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 11, Wednesday, September 24.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 5,Wednesday, September 10.

The Mathematics of Networks Chapter 7. Trees A tree is a graph that –Is connected –Has no circuits Tree.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 6, Friday, September 12.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 4, Monday, September 8.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 20, Monday, October 20.

Discrete Mathematics Lecture 9 Alexander Bukharovich New York University.

Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.

Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley. All rights reserved.

ⅠIntroduction to Set Theory 1. Sets and Subsets

Introduction to Graph Theory

5.4 Shortest-path problem  Let G=(V,E,w) be a weighted connected simple graph, w is a function from edges set E to position real numbers set. We denoted.

CSC 213 – Large Scale Programming. Today’s Goals  Make Britney sad through my color choices  Revisit issue of graph terminology and usage  Subgraphs,

Graph Theory Topics to be covered:

7.1 and 7.2: Spanning Trees. A network is a graph that is connected –The network must be a sub-graph of the original graph (its edges must come from the.

Tree A connected graph that contains no simple circuits is called a tree. Because a tree cannot have a simple circuit, a tree cannot contain multiple.

Aim: Graph Theory - Trees Course: Math Literacy Do Now: Aim: What’s a tree?

 2. Region(face) colourings  Definitions 46: A edge of the graph is called a bridge, if the edge is not in any circuit. A connected planar graph is called.

5.5.2 M inimum spanning trees  Definition 24: A minimum spanning tree in a connected weighted graph is a spanning tree that has the smallest possible.

V Spanning Trees Spanning Trees v Minimum Spanning Trees Minimum Spanning Trees v Kruskal’s Algorithm v Example Example v Planar Graphs Planar Graphs v.

Introduction to Graph Theory

The countable character of uncountable graphs François Laviolette Barbados 2003.

Graph Colouring L09: Oct 10. This Lecture Graph coloring is another important problem in graph theory. It also has many applications, including the famous.

5.5.2 M inimum spanning trees  Definition 24: A minimum spanning tree in a connected weighted graph is a spanning tree that has the smallest possible.

Graph Theory. A branch of math in which graphs are used to solve a problem. It is unlike a Cartesian graph that we used throughout our younger years of.

Trees Dr. Yasir Ali. A graph is called a tree if, and only if, it is circuit-free and connected. A graph is called a forest if, and only if, it is circuit-free.

 Quotient graph  Definition 13: Suppose G(V,E) is a graph and R is a equivalence relation on the set V. We construct the quotient graph G R in the follow.

Graphs Lecture 2. Graphs (1) An undirected graph is a triple (V, E, Y), where V and E are finite sets and Y:E g{X V :| X |=2}. A directed graph or digraph.

Introduction to Graph Theory

Introduction to Graph Theory

CHAPTER 11 TREES INTRODUCTION TO TREES ► A tree is a connected undirected graph with no simple circuit. ► An undirected graph is a tree if and only.

Prims Algorithm for finding a minimum spanning tree

1) Find and label the degree of each vertex in the graph.

Graphs Definition: a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected.

Discrete Mathematics Graph: Planar Graph Yuan Luo

Subgraphs Lecture 4.

Redraw these graphs so that none of the line intersect except at the vertices B C D E F G H.

Discrete Mathematicsq

2.3 Graph Coloring Homework (MATH 310#3W):

The countable character of uncountable graphs François Laviolette Barbados 2003.

5.9.2 Characterizations of Planar Graphs

Graph theory Definitions Trees, cycles, directed graphs.

12. Graphs and Trees 2 Summary

Tucker, Applied Combinatorics, Sec 2.4

Connected Components Minimum Spanning Tree

Trees L Al-zaid Math1101.

Warm Up – Wednesday.

Graph Theory What is a graph?.

Theorem 5.13: For vT‘, l’(v)= min{l(v), l(vk)+w(vk, v)}

Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.

Simple Graphs: Connectedness, Trees

Warm Up – Tuesday Find the critical times for each vertex.

Presentation transcript:

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 10, Monday, September 22

Coloring Edges On the left we see an edge coloring of a graph. The minimum number of colors needed in such a coloring is called the edge chromatic number and is denoted by ’(G).

Theorem 3 (Vizing, 1964) If the maximum degree of a vertex in a graph G is d, then the edge chromatic number of G is either d or d+1. In other words: d · ’(G) · d+1.

Theorem 4 Every planar graph can be 5- colored.

3.1 Properties of Trees Homework (MATH 310#4M): Read 3.2. Do Exercises 3.1: 2,4,6,10,12,14,16,18,24,30 Volunteers: ____________ Problem: 30. On Monday you will also turn in the list of all new terms (marked). On Monday you will also turn in the list of all new terms (marked).

What is a Tree? There are at least three ways to define a tree. We will distinguish the following: tree rooted tree ordered (rooted) tree [will not be used]

A Tree A tree is a connected graph with no circuits. There are several characterizations of trees; compare Theorem 1, p.96 and Exercise 5, p.102. For example: A tree is a connected graph with n vertices and n-1 edges. A tree is a graph with n vertices, n-1 edges and no circuits. A tree is a connected graph in which removal of any edge disconnects the graph. A tree is a graph in which for each pair of vertices u and v there exists an unique path from u to v.

A Spanning Tree Each connected graph has a spanning tree. For finite graphs the proof is easy. [Keep removing edges that belong to some circuit]. For infinite graphs this is not a theorem but an axiom that is equivalent to the renowned axiom of choice from set theory. Note: A spanning subgraph H of G contains all vertices of G.