CSC/Math 4408 and CSC 5408, Applied Graph Theory (Spring 2009)

Syllabus

automatically updated on 26 January 2009

[ Instructor | Class Time and Room | Textbook | Prerequisites | Objectives | Grades and Policies | Schedule| Academic Deadlines (pdf) | Student Page ]

  1. Send a recognizable digital photo of yourself to our webmaster Eduardo from your preferred email address by Wednesday 28 January. His email address is ejbarrios atsign gmail dot com
  2. The first homework, due 9 February, is posted on the Student Page

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Instructor

Dr. Ellen Gethner
Email: ellen dot gethner at cudenver dot edu
Office: North Classroom 2604-A3
Phone: (303) 556 2358
Office hours:

Class Time and Room

Mondays and Wednesdays 4:00-5:15pm in North Classroom 1325.
We will also meet regularly in the Raytheon Lab to do Mathematica work.

Textbook

Graph Theory: A Problem Oriented Approach by Daniel Marcus, published by the Mathematical Association of America (MAA).

If you want to buy other books here are some suggestions. All are available at Amazon.com.

  1. Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, by Sriram Pemmaraju and Steven Skiena; Cambridge University Press, 2003 (for help with the Combinatorica package in Mathematica)
  2. Introduction to Graph Theory (4th Edition), by Robin J. Wilson; Prentice Hall or Addison Wesley, 1996 (for both undergraduates and graduates for general graph theory information)
  3. Introduction to Graph Theory (2nd Edition), by Douglas B. West, Prentice Hall, 2000 (especially for graduate students as a good resource for projects, and most other things graph theoretic)


Other (optional) resources:
  1. Graph Theory with Applications by Bondy and Murty (free download)
  2. Pearls in Graph Theory : A Comprehensive Introduction, by Nora Hartsfield; Dover (available at the Tattered Cover Bookstore in LoDo)
  3. Introduction to Graph Theory, by Chartand and Zhang (2005)
  4. Discrete Mathematics with Algorithms by Albertson and Hutchinson (free download)
  5. Online Graph Theory Warmup by Chris Caldwell
  6. Planar Graph Java Applet Game

Prerequisites

Either CSC 2411 (Discrete Structures) or Math 3000 (Introduction to Abstract Math) or permission of the instructor.

Course Objectives

Grades and Policies

Schedule

Lecture Date Topic(s) Comments Reading Assignment Criteria
One 21 January 2009 (Wednesday) Introduction to Graph Theory The Party Problem; graphs as models Class Notes
Two 26 and 28 January 2009 Vocabulary and Definitions and Mathematica Demo Points, nodes, vertices, endpoints, loops, multiple edges; Directed, Undirected, and Simple Graphs; Multigraph; Real world examples modelled by Graph Theory; Graph Isomorphism with examples; More vocabulary: degree, adjacencies, neighbors, incidence; Classes of Graphs: complete,... To be continued Class Notes Homework 1 handed out on January 26th. Hardcopy due in class on Monday 9 February. Send a copy of your Mathematica work to me by email before class on February 9th.
Three 2 and 4 February 2009 Mathematica Session on Monday; graph classes and some graph gadgets Graph Classes, continued: bipartite, complete bipartite, n-star, path, simple path, cycle, simple cycle; Degree sequence, special property of sum of degrees of a graph, k-regular graphs; Subgraphs (supergraphs): induced, proper, spanning Class Notes and section 1.4 in our text for an overview of Mathematica. Homework 1 is due on Monday: Don't forget to send by email (to Dr. Gethner) your Mathematica notebook that generates your graphic output!
Four 9 and 11 February 2009 More Vocabulary; Special Topic: Graphic Degree Sequences Connecteness: component, maximal, number of connected components,induced subgraph, distance between two vertices, the Girth of a graph, the Peterson Graph. Proof of correctness of the Havel-Hakimi Algorithm. Begin Trees. Class Notes
Five 16 and 18 Feb 2009 Special Topics. Meet in our regular classroom. Matrices and Graphs: New Gadgets; Trees: Definition and Characterization Theorem; powers of the adjacency matrix of a graph and what the entries mean Class Notes and Mathematica and Matrix Multiplication Homework 2 due on 16 February in or before class.
Six 23 and 25 Feb 2009 Feb 16th, 2nd Mathematica Session in the Raytheon Lab; Feb 18th, more on Trees. Adjacency Matrix, continued: an algorithm to test connectivity; Trees: Unique path characterization, number of edges in a tree Class Notes and Section 8.6 of our text, and TreeQ, IsomorphicQ, DegreeSequence, and RealizeDegreeSequence Homework 3 is due on Monday 23 February 2009
Seven 2 and 4 March 2009 Trees and Connectivity Bridges, connectivity and another tree characterization; w(G) versus w(G\e), characterization of bridges; Tree edges are bridges; |E(G)| and and |V(G)| with respect to w(G); Minimum number of edges needed for a graph to be connected; yet another tree characterization theorem (three equivalent statements); Spanning trees and characterization of connected graphs; Special Topic: number of spanning trees by way of deletion and contraction. Class Notes For the midterm, study lecture notes through the Proof of Theorem 3.3. Study homeworks 1-3. You may bring two 4" by 6" notecards with any notes (both sides) that you care to bring to the exam.
Eight 9 and 11 March 2009 Midterm plus Spanning Trees, continued Monday 9th March: Midterm Exam Class notes and Section 8.2.3 in text Homework 4 arrives next week
Nine 16 and 18 March 2009 Spanning Trees, continued Cayley's Theorem on the number of spanning trees of a complete graph; Defn of contraction along an edge; Big Problem in Three Parts that leads to a recursive algorithm to count the number of spanning trees of an arbitrary graph; Big Example of how to use Big Problem to count spanning trees; Matrix Tree Theorem (an alternative method for counting spanning trees) Class notes and Section 8.2.3 again Homework 4 has arrived. It is due on April 1st (no fooling).
null 23-29 March 2009 Spring Break: no classes
Ten 30 March and 1 April 2009 Special Topic in Chapter 3: Optimization by way of finding minimum spanning trees. Kruskal's Algorithm and Prim's Algorithm for finding minimum spanning trees. Definition of weighted graph. Class notes and Section 8.2.2 in text Homework 5 arrives on Wednesday and is due on Wednesday 8 April.
Eleven 6 and 8 April 2009 Chapter 4: Connectivity. A theorem due to Whitney: a relation among edge connectivity, vertex connectivity, and minimum vertex degree. Begin a new special topic related to connectivity: Building reliable communications networks. Given integers k and n with k smaller than n, first find the minimum number of edges that a k-connected graph on n vertices must have. Next, construct such a graph to show that the edge bound is sharp. Class notes and pp 242-243 in text
Twelve 13 and 15 April 2009 Connectivity continued. Chapter 5: Begin planarity. Construct a reliable (ie k-connected graph on n vertices) communications network with the fewest number of edges. Menger's Theorem: connectivity from a different point of view. New Chapter: chapter 5. Definition of plane and planar graphs. Euler's formula for a plane embedding of a planar graph. The complete graph on five vertices is not planar. Class notes and Section 7.2.6 (Harary Graphs) Section 8.7 (planarity) in our text The topic of Hamiltonian and Eulerian Graphs is included in Homework 5, which is due this week on Wednesday.
Thirteen 20 and 22 April 2009 Planarity. Polyhedral Graphs. Average vertex degree of planar (and hence polyhedral) graphs. Dual of a plane graph. Special Topic: The Platonic Solids. Why there are only five of them (tetrahedron, cube, octahedron, icosahedron, dodecahedron). Characterization of Planar Graphs: Subdivision of a graph. Contraction along an edge, revisited. Contractible edge (one that preserves connectivity). Existence of contractible ecdges in simple 3-connected graphs. First half (and harder part) of Kuratowski's Theorem: a graph with no subdivision of K5 or K3,3 is planar. Class notes and knowledge of Mathematica objects TetrahedralGraph, OctahedralGraph, CubicalGraph, DodecahedralGraph, and IcosahedralGraph. Homework 6 (last one!) arrives Monday 14 April and is due on Monday 27 April.
Fourteen 27 and 29 April 2009 Finish Characterization of Planar Graphs. Begin Chapter 6: Coloring Graphs Other half of Kuratowski's Theorem: Any graph with a subdivision of K5 or K3,3 is not planar. New Topic and Chapter: Graph Coloring (an elegant euphemism for scheduling). Crayola Airlines and Their Big Problem. Graph Model: k-coloring, proper k-coloing, color class, chromatic number. Five immediate facts about the chromatic number of a graph (see class notes). Characterization of graphs with chromatic number 2. Brook's Theorem (proof ommitted, technique to come later). Coloring Maps in the Plane: History of the Four Color Theorem. Proof of the Six Color Theorem. Class notes for helpful hints on displaying graphs with colored vertices. ChromaticNumber and MinimumVertexColoring will be useful Mathematica commands to know, too.
Fifteen 4 and 6 May Graph Coloring Finished. Google and Instant Insanity [k-factors, particularly 2-factors of a graph] Proof of the Five Color and Four Color Theorems for Planar Graphs. Thickness-Two Graphs and a Chromatic Mystery; an application to testing printed circuit boards (handouts included). Mathematical Dessert: How Google Works. Games on Graphs: A graph theory solution to the game of Instant Insanity (k-factor of a graph, particularly 2-factors). Class Notes and Herb Wilf's Note on Searching the Web and Grad Student Project by Sergey Brin and Lawrence Page, where Google was Born.
Sixteen Week of 11 May 2009 FINAL EXAM: time and date to be determined by registrar. The Final Exam is comprehensive, but
  1. Google, and
  2. Instant Insanity
will not be on the exam. Mathematica will not be tested, either. 		Extra office hours to be determined You may bring four 4" by 6" notecards with any notes (both sides) that you care to bring to the exam.