4171 - MATH 4171 SPRING 2006 Graph Theory A supplement...

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MATH 4171 SPRING 2006 Graph Theory A supplement Guoli Ding http://math.lsu.edu/ ˜ ding Last Updated: May 3, 2006 Copyright c ± 1996, by Guoli Ding. All Rights Reserved.
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Table of Contents 1. Introduction .............. ............... ............... ................ ............... .......... 1 1.1. Graphs and digraphs 1.2. Examples 1.3. Representations 1.4. Degrees 1.5. Paths and cycles 2. Trees ............... ................ ............... ............... ................ ............... .. 3 2.1. Basic properties 2.2. Minimum cost spanning trees 2.3. Algorithm and Complexity 3. Connectivity ........ ............... ............... ................ ............... ............... .. 6 3.1. Menger theorem 3.2. Maximum fow and minimum cut theorem 3.3. Variations oF Menger theorem 3.4. Connectivity 4. Matchings and independent sets ........... ............... ............... ................ ..... 10 4.1. Bipartite graphs 4.2. Matchings in general graphs 4.2. Independent sets in general graphs 5. Eulerian and hamiltonian graphs and digraphs ...... ................ ............... .......... 13 5.1. Eulerian graphs and digraphs 5.2. Hamiltonian graphs and digraphs 6. Planar graphs ........ ................ ............... ............... ................ ............ 14 6.1. The Euler identity 6.2. Characterizations oF planar graphs 7. Graph colorings ............ ............... ............... ................ ............... ....... 18 7.1. Vertex colorings 7.2. Edge colorings 8. More topics ... ............... ................ ............... ................ ............... ..... 19 8.1. Extremal graph theory 8.2. Counting problems 8.3. Ramsey Theory Index ............. ............... ................ ............... ............... ................ ..... 23 References ....... ............... ................ ............... ................ ............... ...... 24
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1 Introduction 1.1 Graphs and digraphs Let us consider the K¨onigsberg bridges Problem (see page 85). Basically, what Euler did was to consider each land area as an island and thus they can be represented by dots on the plane (which are labeled A,B,C,D ). In addition, bridges are represented by curves on the plane linking the corresponding dots. We need to notice that the resulting diagram is not a geometric object, as dots can be placed on the plane arbitrarily and curves can have any shape. What matters in this diagram are: 1. there are four dots representing the four land areas, 2. there are seven “linkes” representing the seven bridges, and 3. the “links” connect the corresponding dots. Objects with these three characteristics are called graphs. To be precise, a graph is a pair G =( V,E ), where V is a Fnite set whose members are called vertices (they are dots in the above example), and E is a Fnite set whose members are called edges (they are links in the above example), where each edge joins either one or two vertices. The following are related concepts: incident, adjacent, loops, multiple edges, simple graphs, multigraphs, digraphs, arcs ., The word “graph” will mean “multigraph” in this class, unless otherwise stated (in our textbook, “graph” means “simple graph”). 1.2 Examples 1. Street maps and distance maps (weighted graphs and digraphs). 2. Networks (computer, communication, transportation, social). 3. ±riendship graph (edges are not physical links). 4. Tournaments (see page 113). 5. Interval graphs. Let I 1 ,I 2 ,...,I n be intervals on the real line. Then we can deFne a graph with vertex set { 1 , 2 ,...,n } and edge set { ij : I i I j ± = ∅} . Graphs arisen this way are called interval graphs , which were introduced independently by mathematician Haj¨os (1957) and biologist Benzer (1959). See www.bioalgorithms.info/presentations/Ch08 GraphsDNAseq.pdf for applications of interval graphs in biology.
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This note was uploaded on 02/17/2009 for the course MATH 4171 taught by Professor Lax,r during the Spring '08 term at LSU.

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4171 - MATH 4171 SPRING 2006 Graph Theory A supplement...

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