**Record:** Yes

**Level:** Graduate

**Language:** English

**Prerequisite**

**Abstract**

The goal of this course is to give students an overview over the most fundamental concepts and results in modern graph theory.

The topics which we plan to cover include:

Basic notions: graphs, graph isomorphism, adjacency matrix, paths, cycles, connectivity

Trees, spanning trees, Cayley's formula,

Vertex and edge connectivity, 2-connectivity, Mader's theorem, Menger's theorem

Eulerian graphs, Hamilton cycle, Dirac's theorem

Matchings, Hall's theorem, Kőnig's theorem, Tutte's condition

Planar graphs, Euler's formula, basic non-planar graphs, platonic solids

Graph colourings, greedy colourings, Brooks' theorem, 5-colourings of planar graphs, Gallai-Roy theorem

Large girth and large chromatic number, edge colourings, Vizing's theorem, list colourings

Matrix-tree theorem, Cauchy-Binet formula

Hamiltonicity: Chvátal-Erdős theorem, Pósa's lemma, tournaments

Ramsey theory

Turán's theorem, Kővári-Sós-Turán theorem

**Reference**

**Syllabus**

**Lecturer Intro**

Benny Sudakov received his PhD from Tel Aviv University in 1999. He had appointments in Princeton University, the Institute for Advanced Studies and in University of California at Los Angeles. Sudakov is currently professor of mathematics in ETH, Zurich. He is the recipient of a Sloan Fellowship, NSF CAREER Award, Humboldt Research Award, is Fellow of the American Math. Society and was invited speaker at the 2010 International Congress of Mathematicians. He authored more than 300 scientific publications and is on the editorial board of 14 research journals. His main scientific interests are combinatorics and its applications to other areas of mathematics and computer science.

**Lecturer Email:** benny.sudakov@gmail.com

**TA:** Dr. Ruijie Xu, xuruijie@bimsa.cn