The degree of a vertex

The number of edges which start at a given vertex is called the degree of the vertex. On Figure 2 every vertex is labeled with its degree.

  

Figure 2: Degrees of the vertices

There are a few restrictions for the degrees of the vertices in a given graph, see next example as well as Problem 1 of ``More Problems''

Problem 3. In a town there are 15 telephones. Can they be connected by wires so that each telephone is connected with exactly five others?

Solution. Suppose that this is possible. Consider the graph in which the vertices represent telephones, and the edges represent the wires. There are 15 vertices in this graph, and each has degree 5. The number of the edges in the graph must be equal to 15.5/2. But this number is not an integer. So such a graph cannot exist, which means that we cannot connect the telephones as required.

Important part of the last problem is that the number of the odd vertices in any graph must be even (try to prove it). See Homework problem 1.

Problem 4. Can one draw the graph with 4 vertices, each of which is connected by an edge to every other (square and its diagonals) without lifting the pencil from the paper, and tracing over each edge exactly once?

Solution. No. If we can, we will arrive at every vertex as many times as we leave it, with the exception of the initial and terminal vertices. Therefore the degree of each vertex, except for two must be even. Our graph is not the case.

A famous generalization of the last problem is the following:

Theorem. A connected graph can be drawn without lifting the pencil off the paper so that each edge is drawn exactly once if and only if it has no more than two ``odd'' edges (with odd degrees).

It turns out that the proof is not trivial and includes induction with respect to the number of the edges.

A graph in which each vertex is connected by an edge to every other vertex is called complete. So the square with its diagonals is an example of complete graph of 4 vertices.