ELEMENTARY NUMBER THEORY AND MODULAR ARITHMETIC Basic notions of set theory. Natural numbers and the induction principle. Integers. Divisibility and prime numbers. Euler’s function. Congruences. Integers modulo m. Euler’s theorem. Fermat’s little theorem. Elements of group, rings and fields. Finite fields. Chinese remainder theorem. Primality test: Wilson’s theorem and its converse.

ENUMERATIVE COMBINATORICS Cardinality of a set. Basic counting principles. Binomial numbers, Stiefel’s formula and other identities on binomial numbers. Permutations and selections from a set. Binomial theorem. Formal power series. Partial fractions. Generating functions. Linear homogeneous recursions. Fibonacci and Lucas numbers. Closed form of the Fibonacci numbers. Stirling numbers of the second kind. Partitions of a natural number. Ferrer’s diagrams. The Tower of Hanoi problem. Principle of inclusion and exclusion. Formulas of Sylvester and Da Silva. Derangements.

GRAPHS Basic definitions. Planarity of graphs. Euler’s formula. Bipartite graphs. Eulerian graphs. Hamiltonian graphs. Trees. Binary trees. Spanning trees. Vertex colorings. Chromatic number. Edge colorings. Chromatic index. König’s theorem. Matchings in bipartite graphs. Hall’s theorem. Adjacency matrix, incidence matrix with respect to an orientation, Laplacian matrix. Theorem of Poincaré. Spectrum of a graph.