Graphs and Networks MATH20150 Vincent Astier Room: S1.72, Science - PowerPoint PPT Presentation

Graphs and Networks MATH20150 Vincent Astier Room: S1.72, Science South vincent.astier@ucd.ie

How much should I work? ● According to UCD: ● So: 110/12 = 9.16 hours a week ● A bit much, but about 7 hours is possible

How much should I work? ● According to UCD: ● So: 110/12 = 9.16 hours a week ● A bit much, but about 7 hours is possible

How much should I work? ● According to UCD: ● So: 110/12 = 9.16 hours a week ● A bit much, but about 7 hours is possible

How much should I work? ● According to UCD: ● So: 110/12 = 9.16 hours a week ● A bit much, but about 7 hours is possible

How should I work? 1. Go to all lectures and tutorials 2. After each lecture: ● Learn it ● Work on it until you understand everything: You need to be able to do all proofs (except a few), and to explain everything to someone else in the class It could take about 2 hours each week 3. Do all exercises before each tutorial It could take up to 2 hours each week Total: 7 hours a week (can vary)

How should I work? 1. Go to all lectures and tutorials 2. After each lecture: ● Learn it ● Work on it until you understand everything: You need to be able to do all proofs (except a few), and to explain everything to someone else in the class It could take about 2 hours each week 3. Do all exercises before each tutorial It could take up to 2 hours each week Total: 7 hours a week (can vary)

How should I work? 1. Go to all lectures and tutorials 2. After each lecture: ● Learn it ● Work on it until you understand everything: You need to be able to do all proofs (except a few), and to explain everything to someone else in the class It could take about 2 hours each week 3. Do all exercises before each tutorial It could take up to 2 hours each week Total: 7 hours a week (can vary)

How should I work? 1. Go to all lectures and tutorials 2. After each lecture: ● Learn it ● Work on it until you understand everything: You need to be able to do all proofs (except a few), and to explain everything to someone else in the class It could take about 2 hours each week 3. Do all exercises before each tutorial It could take up to 2 hours each week Total: 7 hours a week (can vary)

How should I work? 1. Go to all lectures and tutorials 2. After each lecture: ● Learn it ● Work on it until you understand everything: You need to be able to do all proofs (except a few), and to explain everything to someone else in the class It could take about 2 hours each week 3. Do all exercises before each tutorial It could take up to 2 hours each week Total: up to 7 hours a week (can vary)

Support 1. Myself ● Ask all your questions, at anytime ● Come to my office / email me / etc 2. Tutor ● Again: Ask all your questions 3. Maths Support Centre

Resources ● Course web page: http://maths.ucd.ie/~astier/math20150/ (accessible from my web page) ● It contains: ➔ Course notes (my own handwritten ones) not intended to replace attendance at every lecture ➔ Exercises for tutorials ➔ Recommended books for reading and extra exercises

Graph Theory

Introduction 1. The Königsberg bridge problem Königsberg, 1736 (now Kaliningrad, Russia)

Map of the bridges: Question: Is it possible to start at one point, cross every bridge exactly once, and end up at the same point?

Solution (and invention of graph theory): Leonhard Euler, 1736 ● Represent it as a “graph”: ● Answer: Impossible ● Euler’s answer covers all such questions, for instance:

Solution (and invention of graph theory): Leonhard Euler, 1736 ● Represent it as a “graph”: ● Answer: Impossible ● Euler’s answer covers all such questions, for instance:

2. The 4 colour problem (1852) Can any map be coloured such that any two adjacent countries are of different colours? Translation into graphs: Each country become a dot Join dots if they are adjacent: The problem becomes: Can we colour the points with 4 colours such that adjacent points have different colours?

2. The 4 colour problem (1852) Can any map be coloured such that any two adjacent countries are of different colours? Translation into graphs: Each country become a dot Join dots if they are adjacent: The problem becomes: Can we colour the points with 4 colours such that adjacent points have different colours?

2. The 4 colour problem (1852) Can any map be coloured such that any two adjacent countries are of different colours? Translation into graphs: Each country become a dot Join dots if they are adjacent: The problem becomes: Can we colour the dots with 4 colours such that connected dots have different colours?

Answer (Appel, Haken, 1976): Yes The proof used computers to examine some 15000 special cases. (Grear achievement, but the use of computers is not without problems) Still no proof that does not involve computers