# Math 38. Graph Theory

Spring 2008

• Lectures:           MWF 12:30-1:35 in Haldeman 028

• X-hour:           Tu 1:00-1:50

• Instructor:         Peter Winkler

• Office Hours:    M 2-3, W 5-6, F 10-11

• Office:               Kemeny 231

• Email:

• Phone:               646-3468

Announcements

The final exam for Math 38 will be in our regular classroom, Haldeman 028. It's at 3pm Sunday June 1; you'll have 3 hours to complete the exam, but it's actually designed to take only two.

Course description

This course will cover the fundamental concepts of Graph Theory: simple graphs, digraphs, Eulerian and Hamiltonian graphs, trees, matchings, networks, paths and cycles, graph colorings, and planar graphs. Famous problems in Graph Theory include: Minimum Connector Problem (building roads at minimum cost), the Marriage Problem (matching men and women into compatible pairs), the Assignment Problem (filling n jobs in the best way), the Network Flow Problem (maximizing flow in a network), the Committee Scheduling Problem (using the fewest time slots), the Four Color Problem (coloring maps with four colors so that adjacent regions have different colors), and the Traveling Salesman Problem (visiting n cities with minimum cost).

Textbook

Introduction to Graph Theory by Douglas B. West, Second edition (available at Wheelock Books).

Tentative syllabus

This is the syllabus we actually covered.

 Chapters Brief Description Week 1 1.1 Definitions; bipartite graphs; chromatic number; adjacency matrix; isomorphism; decomposition; connectedness; subgraphs and induced subgraphs Week 2 1.2, 1.3 Path versus walk; cycles; cut-edge and cut-vertex; TONCAS for Eulerian property; sum of degrees; extremality; induction Week 3 1.4 Paths; cycles; strong digraphs; Eulerian digraphs Week 4 2.1, 2.2 Equivalent definitions of tree; distance and diameter; Cayley's Theorem and Pruefer code; recurrence for counting spanning trees; matrix-tree theorem (not proof); BEST Theorem and proof; graceful tree conjecture Week 5 2.3 Kruskall's algorithm and reverse version; Dijkstra's algorithm Week 6 3.1 Maximal, maximum and perfect matchings; augmenting paths; Hall's marriage theorem; min-max theorems: vertex cover versus matchings; Koenig's Theorem on edge-coloring a bipartite multigraph [not in text] Week 7 4.1, 4.2, 4.3 Edge-connectivity; blocks; k-connectedness; 2-connected graphs; connectivity in digraphs; Menger's Theorem (not proof); line graphs; feasible and maximal flows; min flow/max cut theorem (not proof) Week 8 5.1, 5.2 Cartesian product; interval graphs; Brooks' Theorem; Mycielski's construction; color-criticality; Turan's Theorem and proof Week 9 6.1, 6.2 Planar graphs versus plane graphs; non-planarity of K5 and K3,3; faces and the dual graph; Euler's formula; bound on number of edges; maximal planar graphs; Kuratowski's TONCAS (not proof); list coloring and Thomassen's Theorem [not in text] Week 10 7.2 Necessary and sufficient conditions for Hamiltonicity; Dirac's Theorem; Gray code [not in text]; mid-levels problem[not in text]