Given a network, begin at some vertex and travel on each edge exactly once, and then return to the
starting vertex. Such as path is called an Euler circuit. There
is a solution to the question of whether an Euler circuit for a given network exists; it is
known as Euler's circuit theorem.
Every vertex on a graph with an Euler circuit has an even degree, and conversely, if in a
connected graph every vertex has an even degree, then the graph has an Euler circuit.
Given a network, begin a some vertex and travel to each vertex exactly once, ending at the original
vertex. Such a path is called a Hamiltonian cycle. There is no solution to the question of whether
a Hamiltonian cycle for a given network exists. The methods we use in this text are brute force
(listing all possible routes); nearest neighbor (at each city, go to the nearest neighbor next);
and sorted-edge (sometimes the nearest neighbor plan will form a loop without going to some city)
procedures.
Draw a graph showing the cities and the distances; identify the starting vertex.
Step 1: Choose the edge attached to the starting vertex that has the shortest
distance or the lowest cost. Travel along this edge to the next vertex.
Step 2: At the second vertex, travel along the edge with the shortest distance
or lowest cost. Do not choose a vertex that would lead to a vertex already visited.
Step 3: Continue until all vertices are visited until arriving back at the
original vertex.