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, 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.
A class has more than 32, but less than 40 people. Every boy is friends with three girls, and every girl is friends with five boys. How many people are there in the class?
In the oriented graph, there are 101 vertices. For each vertex, the number of ingoing and outgoing edges is 40. Prove that from each vertex you can get to any other, having gone along no more than three edges.
In a country, each two cities are connected with a one-way road.
Prove that there is a city from which you can drive to any other whilst travelling along no more than two roads.
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).
Prove that in a bipartite planar graph \(E \geq 2F\), if \(E \geq 2\) (\(E\) is the number of edges, \(F\) is the number of regions).
Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).
Is the number 12345678926 square?