## Graph Theory (Fall 2016)

Lecturer: Wojciech Samotij
e-mail: samotij(at)post.tau.ac.il
Course #: 0366-3267-01
Sunday, 15:10–18:00
Shenkar Physics 222

Office hours:
Tuesday, 16:00–17:00
Schreiber 002

### Course description

Among topics that will be covered in the class are the following: graphs and subgraphs, trees, connectivity, Euler tours, Hamilton cycles, matchings, Hall's theorem and Tutte's theorem, edge coloring and Vizing's Theorem, independent sets, Turán's theorem and Ramsey's theorem, vertex coloring, planar graphs, directed graphs, probabilistic methods and linear algebra tools in graph theory.

Prerequisite courses: Discrete mathematics or Introduction to combinatorics and graph theory, Linear algebra, and Introduction to probability.

Homework exercises will be given during the course and will account for 10% of the final grade. There will also be a final exam.

The course will be taught in English.

### Textbooks

B. Bollobás, Modern Graph Theory, Springer, 2002
J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008
R. Diestel, Graph Theory, Springer, 2012 (4th edition)
D. B. West, Introduction to Graph Theory, Pearson Prentice Hall, 2001 (2nd edition)

M. Aigner and G. Ziegler, Proofs from THE BOOK, Springer, 2014 (5th edition) N. Alon and J. H. Spencer, The Probabilistic Method, Wiley, 2008 (3rd edition)
L. Lovász, Combinatorial problems and exercises, AMS Chelsea Publishing, 2007 (2nd edition)

### Course outline

October 30
basic definitions; graph isomorphism, labeled and unlabeled graphs; the adjacency and the incidence matrices; subgraphs and induced subgraphs; the complement and the line graph of a graph; complete and empty graphs, cliques and independent sets; bipartite graphs; vertex degrees; degree sum formula and the handshaking lemma; walks, trails, paths and cycles, girth and circumference; every u,v-walk contains a u,v-path; the longest path and cycle in a graph of a given minimum degree; connectivity and connected components
November 6
Sperner's lemma (in dimension 2); Brouwer's fixed point theorem (in dimension 2); trees and forests; basic properties of trees: every nontrivial tree contains at least two leaves, deleting a leaf from a tree yields a smaller tree; four equivalent definitions of a tree; cut-edges; Cayley's formula for the number of labeled trees with n vertices (with proof)
November 13
edge contraction and the contraction–deletion recursive formula for the number of spanning trees; Kirchhoff's maxtrix tree theorem: the statement and derivation of Cayley's formula; a proof of Kirchhoff's matrix tree theorem; the Cauchy–Binet formula; connectivity of a graph; highly connected subgraphs in graphs of large average degree (Mader's theorem)
November 20
edge-connectivity; κ(G) ≤ κ'(G) ≤ δ(G); structural characterization of 2-connected graphs ("ear decomposition"); blocks and block-decompositions; flows in directed graphs; the max-flow min-cut theorem; the integrality theorem; flows in directed graphs with vertex capacity bounds
November 27
the max-flow min-cut theorem for vertex capacity bounds; Menger's theorem; global version of Menger's theorem; the Fan Lemma; Eulerian circuits; the characterisation of Eulerian multigraphs; Hamilton cycles; Dirac's theorem; Ore's theorem
December 4
The Chvátal–Erdős theorem; matchings, factors, and vertex covers; Hall's marriage theorem and corollaries: every nonempty regular bipartite graph has a perfect matching, every regular graph with positive even degree has a 2-factor (Petersen's theorem); systems of distinct representatives; König's theorem; Tutte's matching theorem
December 11
Proof of Tutte's matching theorem; every bridgeless 3-regular graph has a perfect matching; vertex coloring; relations between the clique number, the chromatic number, and the independence number; the Nordhaus–Gaddum theorem; the greedy coloring algorithm; degeneracy of a graph; Brooks' theorem
December 18
CLASS CANCELLED
December 25
NO CLASS (Hanukkah)
January 1
Proof of Brooks' theorem; color-critical graphs; color-critical graphs have no cutvertices; every (k+1)-critical graph is k-edge-connected (Dirac's theorem); a construction of triangle-free graphs with large chromatic number; graphs with large chromatic number and no short cycles; edge coloring
January 8
König's theorem (χ'(G) = Δ(G) for bipartite G); Vizing's theorem (χ'(G) ≤ Δ(G)+1); Ramsey numbers; the upper bound on R(s,t) of Erdős and Szekeres; determination of R(3,3), R(3,4), and R(4,4)
January 15
A probabilistic lower bound on the diagonal Ramsey numbers R(s,s); multicolour Ramsey numbers; Schur's theorem; Turán numbers; Mantel's theorem; Turán graphs; Turán's theorem (with 2 proofs)
January 22
Zarankiewicz's problem and the Kővari–Sós–Turán theorem; a construction of large K2,2-free graphs based on point-line incidences in projective planes; the Erdős–Stone theorem (without proof); planar graphs; the Jordan Curve Theorem and the Jordan–Schönfliess theorem (without proofs); K5 is not planar; subdivisions and minors; Kuratowski's theorem (without proof)
January 29
Wagner's theorem (without proof); faces of a plane graph; Whitney's theorem (faces of 2-connected plane graphs are cycles); the plane dual G* of a plane graph G; duals are connected; (G*)* = G for connected G; duality of edge deletion and edge contraction; Euler's formula; an upper bound on the number of edges of a simple planar graph; K3,3 and K5 are not planar via Euler's formula; the four colour problem; Heawood's theorem (planar graphs are 5-colourable)

### Homework

Assignment #1 (due on November 20)
Assignment #2 (due on December 11)
Assignment #3 (due on January 1)
Assignment #4 (due on January 29)

### Exam

The final exam will take place on Sunday, the 26th of February at 9:00; the second attempt is scheduled for Monday, the 14th of August at 9:00. Below is a list of theorems that were presented in class and whose proofs one is expected to know during the final exam. Please remember that your grades from the homework assignments account for 10% of the final grade.

List of theorems for the final exam