Combinatorics Through Guided Discovery, 2004a

42 2. Applications of Induction and Recursion in Combinatorics and Graph Theory
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Figure 2.3.1: Three different graphs
Each gray circle in the figure represents a vertex; each line segment represents
an edge. You will note that we labelled the vertices; these labels are names we chose
to give the vertices. We can choose names or not as we please. The third graph
also shows that it is possible to have an edge that connects a vertex (like the one
labelled y) to itself or it is possible to have two or more edges (like those between
vertices v and y) between two vertices. The degree of a vertex is the number of
times it appears as the endpoint of edges; thus the degree of y in the third graph
in the figure is four.
◦ Problem 100. In the graph on the left in Figure 2.3.1, what is the degree of
each vertex?
◦ Problem 101. For each graph in Figure 2.3.1 is the number of vertices of
odd degree even or odd?
⇒ ·
Problem 102. The sum of the degrees of the vertices of a (finite) graph is
related in a natural way to the number of edges.
(a) What is the relationship? (h)
(b) Find a proof that what you say is correct that uses induction on the
number of edges. Hint: To make your inductive step, think about
what happens to a graph if you delete an edge. (h)