Assignment: Euler and Hamiltonian Circuits

  1. You and your friends want to tour the southwest by car. You will visit the nine states below, with the following rather odd rule: you must cross each border between neighboring states exactly once (so, for example, you must cross the Colorado-Utah border exactly once). Can you do it? If so, does it matter where you start your road trip? What fact about graph theory solves this problem?
    Map of the US with the following states shaded: California, Arizona, New Mexico, Texas, Oklahoma, Nebraska, Colorado, Utah, Nevada. Each of the states shares a border with another of the states listed.
  2. Below is a graph representing friendships between a group of students (each vertex is a student and each edge is a friendship). Is it possible for the students to sit around a round table in such a way that every student sits between two friends? What does this question have to do with paths?