Section 6.2: Euler Paths and Hamiltonian Circuits

Abstract:

Graphs are useful for characterizing two interesting problems: the traveling salesman problem, and the highway inspector problem. The problem in each case is to traverse a network in an optimal way, whether the focus is on the nodes (Sir William Rowan Hamilton, 1805-1865) or the arcs (Leonhard Euler, 1707-1783).

Euler Paths

Definition: an Euler Path is a path in which each arc is used exactly once.

Euler got interested in these arcs when he encountered the Königsberg bridge problem (p. 421); a game in which the object was to cross every bridge without crossing any bridge twice. Euler solved this problem by inventing and then using Graph Theory, basically!

The bridges are the arcs, and the land masses are nodes, turned into the graph of Figure 6.5, p. 421.

To do: Practice 7, p. 421

Theorem: in any graph, the number of odd nodes (nodes of odd degree) is even.

Outline of proof:

1. Suppose that there are A arcs, and N nodes. Each arc contributes 2 ends; the number of ends is 2A, and the degrees satisfy

   

2. 

   and the left hand side is even (call it 2m).

3. The sum of two odd degrees is even, so assume (we proceed by contradiction) that there is an odd number of odd nodes. We can pair up all but one (say i=k), and then

   

4. From which we conclude that

   

   which means that was, in fact, even; but this is a contradiction. Hence, the number of odd nodes is even.

Theorem: an Euler path exists in a connected graph there are either two or zero odd nodes.

• Is this obvious? Why only two odd nodes?
• The two odd case reduces to the even case: start at one odd node, and trace a path to the other. Remove this subgraph, and what's left (and what might that be?) has even nodes only; so, since an Euler path exists for even noded graphs, we can reattach the pieces to form the original graph, with its Euler path.

To do: Practice 9, p. 423

EulerPath Algorithm

The EulerPath algorithm makes use of the adjacency matrix representation of a graph to check for Euler paths. It simply counts up elements in a row i of the matrix (the degree of node i), and checks whether that's even or odd; if in the end there are not zero or two even nodes, there's no Euler path!

To do: Exercise 11, p. 427

Hamiltonian Circuit Problem (the traveling salesman)

Definition: a Hamiltonian Circuit is a cycle using every node of the graph.

An example is a complete graph, like : there is a path from each node to every other node, so no matter where you start, you can trace a cycle through every node (breaks down for .

To do: Practice 11, p. 425

To do: Exercise 14, p. 427 (using trees, symmetry, and exhaustion!)

Unfortunately, there's no nice HamiltonCircuit algorithm for determining when there is a circuit (only very grungy ones!).