1 MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 7, Monday, September 15
2 Chapter 1 Review Homework (MATH 310#3M): Read Supplement I (pp 46-48). Write down a list of all newly introduced terms (printed in boldface or italic)Do Ch.1 Supplement II. Exercises : 4,14,16,20,28,30,36.Volunteers:____________Problem: 30.
3 Example 5: Gray Code Example: n = 3. There are 8 binary sequences: 000001010011100101110111There are 2n binary sequences of length n.An ordering of 2n binary sequences with the property that any two consecutive elements differ in exactly one position is called a Gray code.
4 HypercubeThe graph with one vertex for each n-digit binary sequence and an edge joining vertices that correspond to sequences that differ in just one position is called an n-dimensional cube or hypercube.v = 2ne = n 2n-1
6 4-dimensional Cube and a Famous Painting by Salvador Dali Salvador Dali (1904 – 1998) produced in 1954 the Crucifixion (Metropolitan Museum of Art, New York) in which the cross is a 3-dimensional net of a 4-dimensional hypercube.
7 Gray Code - Revisited010011A Hamilton path in the hypercube produces a Gray code.111110100101000001
8 2.3 Graph Coloring Homework (MATH 310#3W): Read 2.4. Write down a list of all newly introduced terms (printed in boldface or italic)Do Exercises 2.3: 2,4,6,7,12,14,18Volunteers:____________Problem: 7.On Monday you will also turn in the list of all new terms with the following marks+ if you think you do not need the definition on your cheat sheet,check (if you need just the term as a reminder),- if you need more than just the definition to understand the term.
9 Coloring and Chromatic Number A coloring of a graph G assigns colors to the vertices of G so that adjacent vertices are given different colors.The minimal number of colors required to color a given graph is called the chromatic number of a graph.
10 Example 1: Simple Graph Coloring Find the chromatic number of the graph on the left.