NL-NWO 1965 Problem 4

We consider a number of points in a plane. Each of these points is connected to at least one of the other points by a line segment, in such a way that a figure arises that does not break up into different parts (that is, from any point along drawn line segments we can reach any other point).. We assign a point the "order" $n$ , when in this point $n$ line segments meet. We characterize the obtained figure by writing down the order of each of its points one after the other. For example, a hexagon is characterized by the combination $\{2,2,2,2,2,2\}$ and a star with six rays by $\{6,1,1,1,1,1,1\}$ . (a) Sketch a figure' belonging to the combination $\{4,3,3,3,3\}$ . (b) Give the combinations of all possible figures, of which the sum of the order numbers is equal to $6$ . (c) Prove that every such combination contains an even number of odd numbers.