CSCI 4408/5408, Applied Graph Theory (Spring 2018)

Professor Ellen Gethner

Syllabus


automatically updated on 8 January 2018

[ Instructor and Office Hours | Class Time and Room | Textbook | Prerequisites | Objectives | Grades and Policies | Schedule| Academic Deadlines | Student Page (not yet available) ]




Instructor

Dr. Ellen Gethner
Email: ellen dot gethner at ucdenver dot edu
Office: Lawrence Street Center LW 817
Phone: 303 315 1405
Office hours (make an appointment by calling the CS office at 303-315-1408):

Class Time and Room

Tuesdays and Thursdays 2:00-3:15pm in Lawrence Street Center Room 844

Textbook

Graph Theory: A Problem Oriented Approach by Daniel Marcus, published by the Mathematical Association of America (MAA). Available at the Auraria book store and many other places.

Other Resources

  1. 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)
  2. 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)
  3. Pearls in Graph Theory : A Comprehensive Introduction, by Nora Hartsfield; Dover
  4. Introduction to Graph Theory, by Chartand and Zhang (2005)
  5. Discrete Mathematics with Algorithms by Albertson and Hutchinson (free download)
  6. Online Graph Theory Warmup by Chris Caldwell
  7. Planar Graph Java Applet Game

Prerequisites

CSCI 2511 (Discrete Structures)

Course Objectives

Grades and Policies

Schedule

Lecture Date Topic(s) Comments Reading Assignment and Quiz Schedule
One 16 and 18 January Introduction to Graph Theory The Party Problem; graphs as models; Vocabulary and Definitions; Mathematica demo (probably) on the 18th Class Notes and from our textbook: the preface and Chapters A and B Homework 1 e-mailed to students
Two 23 and 25 January Vocabulary and Definitions, continued 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 and from our textbook: the preface and Chapters A and B
Three 30 Jan and 1 February Mathematica Session on Thursday; 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 Chapters C and D in our text. Quiz 1 on Tuesday; Homework 2 e-mailed to students
Four 6 and 8 February 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. An open research question. Begin Trees. Class Notes
Five 13 and 15 Feb Special Topics. 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 Quiz 2 on Tuesday; Homework 3 e-mailed to students
Six 20 and 22 Feb 2nd Mathematica Session on Thursday; Adjacency Matrix, continued: an algorithm to test connectivity; Trees: Unique path characterization, number of edges in a tree Class Notes and Sections A and D of our text, and TreeQ, IsomorphicQ, DegreeSequence, and RealizeDegreeSequence Quiz 3 is on Thursday; Homework 4 e-mailed to students
Seven 27 Feb and 1 March 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
Eight 6 and 8 March Midterm plus Spanning Trees, continued Tuesday 6th March: Exam One For the midterm, study lecture notes through Theorem 3.14 (Matrix Tree Theorem). Study homeworks and quizzes 1-3, and read relevant sections of the textbook, and do problems from those sections as well.
Nine 13 and 15 March 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 D again
null 19-23 March Spring Break: no classes
Ten 27 and 29 March Special Topic: 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 E in text Quiz 4 on Tuesday; Homework 5 e-mailed to students
Eleven 3 and 5 April 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
Twelve 10 and 12 April Connectivity continued. Section H: 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. 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 (Harary Graphs and more on planarity) Quiz 5 on Tuesday; Homework 6 e-mailed to students
Thirteen 17 and 19 April 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 edges 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.
Fourteen 24 and 26 April Finish Characterization of Planar Graphs. Begin Section K: 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. Quiz 6 on Tuesday
Fifteen 1 and 3 May Exam plus graduate student presentations Tuesday 1 May, Exam 2; Graduate Student Presentations on Thursday Exam 2 is comprehensive.
Sixteen Week of 9 May 2016 Graduate Student Presentations during our final exam time if necessary Extra office hours and graduate student research paper presentations schedule to be determined