Prove that there exists a graph with 2n vertices whose degrees are \(1, 1, 2, 2, \dots , n, n\).
a) they have 10 vertices, the degree of each of which is equal to 9?
b) they have 8 vertices, the degree of each of which is equal to 3?
c) are they connected, without cycles and contain 6 edges?
Prove that a graph, in which every two vertices are connected by exactly one simple path, is a tree.
Prove that, in a tree, every two vertices are connected by exactly one simple path.
Prove that there is a vertex in the tree from which exactly one edge emerges (such a vertex is called a hanging top).
In a graph, all the vertices have degree of 3. Prove that there is a cycle in it.
There are seven lakes in some country, connected by ten non-overlapping canals, and each lake can be reached from any other. How many islands are there in this country?
Prove that for a flat graph the inequality \(2E \geq 3F\) is valid.
On the plane 100 circles are given, which make up a connected figure (that is, not falling apart into pieces). Prove that this figure can be drawn without taking the pencil off of the paper and going over any line twice.
At a conference there are 50 scientists, each of whom knows at least 25 other scientists at the conference. Prove that is possible to seat four of them at a round table so that everyone is sitting next to people they know.