## Combinatorics

## Members

Profa. Ana Shirley Silva

Prof. Fabrício Siqueira Benevides

Prof. Julio Cesar Silva Araujo

## Description

Combinatorics is one of the most recent areas of mathematics. It concerns the study of finite or countable discrete structures in its many aspects. It is an attractive area for it has a small amount of previous knowledge for a student to understand the statements of fundamental results. However, it requires a lot of maturity to appreciate the solution of such problems and to solve modern ones. Recently, the area has been growing steadily and has attracted a lot of attention nationally and internationally, for example, through its applications and interaction with other areas.

Some branches of Combinatorics include: Graph Theory, Extremal combinatorics, Probabilistic combinatorics, Polyhedral combinatorics, Additive combinatorics, Enumerative combinatorics, Order theory, Matroid theory, Algebraic combinatorics, Analytic combinatorics, etc. From all these branches, our program focus on the first tree.

Graph Theory is probably the oldest topic in the area, dating back to 1736 when Euler solved the problem about the Konisgberg’s bridges. Nowadays, graphs are a fundamental structure in combinatorics. Our research in Graph Theory consists mainly in the study of structural properties of certain classes of graphs and the study of optimization problems in graphs, with emphasis on colorings/partition problems.

Extremal combinatorics study the relationships between various numerical and structural parameters of discrete structures. In general, it answers question of the type: how large or how small a certain parameter must be, given that a condition about another parameter must hold. Among the classic topics, we are interested, for example, in: Ramsey theory, extremal set theory and colorings.

Probabilistic combinatorics, uses methods of probability as a mean to obtain deterministic results about discrete structures. Here, it is worth mentioning the study of Random Graphs, Percolation and Bootstrap Percolation.

Combinatorics was included as a research area in our program only in 2011, when two specialists where hired and a professor from the Computer Science Department (DCC-UFC), Prof. Cláudia Linhares Sales, has joined the program. We have deep collaboration with the research group ParGO (http://dgp.cnpq.br/buscaoperacional/detalhegrupo.jsp?grupo=0089103FAEPUHI) which has its headquarters at DCC-UFC.