There are 30 people, among which some are friends. Prove that the number of people who have an odd number of friends is even.
In some country 89 roads emerge from the capital, from the city of Dalny – one road, from the remaining 1988 cities – 20 roads (in each).
Prove that from the capital you can drive to Dalny.
25 cells were coloured in on a sheet of squared paper. Can each of them have an odd number of coloured in neighbouring cells?
Can the degrees of vertices in the graph be equal to:
a) 8, 6, 5, 4, 4, 3, 2, 2?
b) 7, 7, 6, 5, 4, 2, 2, 1?
c) 6, 6, 6, 5, 5, 3, 2, 2?
In the graph, each vertex is either blue or green. Each blue vertex is linked to five blue and ten green vertices, and each green vertex is linked to nine blue and six green vertices. Which vertices are there more of – blue or green ones?
In a graph there are 100 vertices, and the degree of each of them is not less than 50. Prove that the graph is connected.
In a graph, three edges emerge from each vertex. Can there be a 1990 edges in this graph?
Prove that the number of US states with an odd number of neighbours is even.
The faces of a polyhedron are coloured in two colours so that the neighbouring faces are of different colours. It is known that all of the faces except for one have a number of edges that is a multiple of 3. Prove that this one face has a multiple of 3 edges.
a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?
b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).