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 colouring and Vizing's Theorem, independent sets, Turán's theorem and Ramsey's theorem, vertex colouring, 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.
The course will be taught in English.
B. Bollobás, Modern Graph Theory
, Springer, 2002
J. A. Bondy and U. S. R. Murty, Graph Theory
, Springer, 2008
R. Diestel, Graph Theory
, Springer, 2016 (5th edition) – freely available for online viewing at diestel-graph-theory.com
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, 2016 (4th edition)
L. Lovász, Combinatorial problems and exercises, AMS Chelsea Publishing, 2007 (2nd edition)
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; 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; edge contraction and the contraction–deletion recursive formula for the number of spanning trees; Cayley's formula for the number of labeled trees with n vertices
Proof of Cayley's formula; Kirchhoff's maxtrix tree theorem: the statement and derivation of Cayley's formula; 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); edge-connectivity; κ(G) ≤ κ'(G) ≤ δ(G); structural characterisation 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; 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; 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);
König's theorem; Tutte's matching theorem
Every bridgeless 3-regular graph has a perfect matching; vertex colouring; relations between the clique number, the chromatic number, and the independence number; the Nordhaus–Gaddum theorem; the greedy colouring algorithm; degeneracy of a graph; Brooks' theorem
NO CLASS (Hanukkah)
Colour-critical graphs; colour-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 colouring; 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; 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); Zarankiewicz's problem and the Kővari–Sós–Turán theorem; a construction of large K2,2
-free graphs based on point-line incidences; 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); 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
are not planar via Euler's formula; the four colour problem; Heawood's theorem (planar graphs are 5-colourable); sketch of the proof of Wagner's theorem
The final exam will take place on Sunday, the 10th of February at 9:00; the second attempt is scheduled for Friday, the 9th 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.
List of theorems for the final exam