  Our goal is to prove new results in graph theory and combinatorics thanks to
the speed of computers, used with smart algorithms. We tackle four problems.
  The four-colour theorem states that any map whose countries are connected can
be coloured with 4 colours such that neighbouring countries have differnt
colours. It was the first result proved using computers, in 1989. We wished to
automatise further its proof. We explain the proof and provide a program which
replays the proof. It also makes it possible to obtain other results with the
same method. We give ideas to automatise the search for discharging rules.
  We also study the problems of domination in grids. The simplest one is the
one of domination. It consists in putting a stone on some cells of a grid such
that every cell has a stone, or has a neighbour with a stone. This problem was
solved in 2011, using computers to prove a formula giving the minimum number of
stones needed. We adapt this method for the first time for variants of the
domination. We solve partially two other problems and give for them lower
bounds for grids of arbitrary size.
  We also tackle the counting problem for dominating sets. How many dominating
sets are there for a given grid? We study this counting problem for the
domination and three variants. For each of these problems, we prove the
existence of asymptotic growths rates for which we give bounds.
  Finally we study polyominoes and the way they can tile rectangles. We tried
to solve a problem from 1989: is there a polyomino of odd order? It consists in
finding a polyomino which can tile a rectangle with an odd number of copies,
but cannot tile any smaller rectangle. We did not manage to solve this problem,
but we made a program to enumerate polyominoes and try to find their orders,
discarding those which cannot tile rectangles. We also give statistics on the
orders of polyominoes of size up to 18.
