Learning objectives

This course provides students with the basic concepts of Graph Theory and the basics of Discrete and Computational Geometry. At the end of the course, the student will know the main classical theorems of graph theory, in particular about structural properties, colorings, matchings, embeddings and flow problems. He/she will also be familiar with basic Discrete Geometry results and with some classical algorithms of Computational Geometry. He/she will have the perception of links with some problems in non mathematical contexts. he/she will be able to produce rigorous proofs on all these topics and he/she will be able to read articles and texts of Graph Theory and Discrete Geometry.

Prerequisites and basic notions

Fundamental idea of general topology, affine and euclidean geometry.

Program

GRAPH THEORY
-Definitions and basic properties.
-Matching in bipartite graphs: Konig Theorem and Hall Theorem. Matching in general graphs: Tutte Theorem. Petersen Theorem.
-Connectivity: Menger's theorems.
-Planar Graphs: Euler's Formula, Kuratowski's Theorem.
-Colorings Maps: Four Colours Theorem, Five Colours Theorem, Brooks Theorem, Vizing Theorem.

DISCRETE GEOMETRY
-Convexity, convex sets convex combinations, separation. Radon's lemma. Helly's Theorem.
-Lattices, Minkowski's Theorem, General Lattices.
-Convex independent subsets, Erdos-Szekeres Theorem.
-Intersection patterns of Convex Sets, the fractional Helly Theorem, the colorful Caratheodory theorem.
-Discrete surfaces and discrete curvatures.

COMPUTATIONAL GEOMETRY
-General overview: reporting vs counting.
-Convex-hull problem: Graham's scan and other algorithms.
-Polygons and Art Gallery problem. Art Gallery Theorem, polygon triangulation.
- Voronoi diagram and Fortune's algorithm.
- Delaunay triangulation properties and Minimum spanning tree.

Bibliography

Didactic methods

Lectures and exercise sessions.

Learning assessment procedures

Written test (2 hours).
The written exam on Graph Theory consists of three/four exercises and two questions (1 on general definition / concepts and 1 with a proof of a theorem presented during the lectures).

Oral Test (Mandatory)
It is a discussion with the lecturer on definitions and proofs discussed during the lectures about Discrete and Computational Geometry.

Evaluation criteria

To pass the exam, students must show that:
- they know and understand the fundamental concepts of graph theory
- they know and understand the fundamental concepts of Discrete and Computational Geometry
- they have analysis and abstraction abilities
- they can apply this knowledge in order to solve problems and exercises and they can rigorously support their arguments.

Criteria for the composition of the final grade

Written exam maximum 30/30. Oral part, if positive, could add at most 5 points.

Exam language

English