The lecture notes give two examples of voter preferences for three candidates, A, B, and C. In the first example, C was the Condorcet winner, the Borda winner, and the widest path winner. In the second example, there was no Condorcet winner. Find another example of voter preferences for these three candidates, in which there is a Condorcet winner, but in which the Borda winner is different from the Condorcet winner. (Hint: Have the Condorcet winner be someone whom most voters either love or hate, and then unbalance the remaining voter choices to make someone else be the Borda winner.)
You already know that the widest path method should pick the Condorcet winner from your answer to question 1. Draw the graph of voter preferences used in applying the widest path method to this example. For each ordered pair of vertices, state the width of the widest path connecting them.
Describe an infinite graph in which all vertices have finite degree and exactly one vertex has odd degree.
Describe an algorithm for the following problem: You are given as input a finite connected undirected graph in which all vertices have even degree. You should choose and output a direction for each edge, producing a strongly connected directed graph in which each vertex has equal numbers of incoming and outgoing edges. (Hint: Start by finding an Euler tour. You do not need to describe how to find this tour.)