Coloring Vertex Degree

20 phones are connected by wires in such a way that each wire connects two phones, each pair of phones is connected by no more than one wire, and no more than two wires extend from each phone. It is necessary to color the wires (each wire entirely in one color) so that the wires extending from each phone are of different colors. What is the smallest number of colors sufficient for such a coloring?
We will represent phones as points and the wires connecting them as segments connecting these points. Since no more than two segments come out of each point, the resulting graph breaks down into open paths and simple cycles. Note that an open path can be painted with two colors, and a simple cycle with three. For this, it is enough to paint every other segment in an open path, and in a cycle, paint one segment in color 3, and the remaining ones alternately in colors 1 and 2. Additionally, note that if there is a triangle in the resulting graph, two colors will not be enough. Therefore, the number of colors needed is 3.

## Answer

Three colors.