• Euler's Theorems are examples of existence theorems
• existence theorems tell whether or not something exists (e.g. Euler circuit)
• but doesn't tell us how to create it!
• we want a constructive method for finding Euler paths and circuits
• methods (well-defined procedures, recipes) for construction are called algorithms
• there is an algorithm for constructing an Euler circuit:  Fleury's algorithm
• we will illustrate it with the graph:

1.  pick any vertex to start
2.  from that vertex pick an edge to traverse (see below for important rule)
3.  darken that edge, as a reminder that you can't traverse it again
4.  travel that edge, coming to the next vertex
5.  repeat 2-4 until all edges have been traversed, and you are back at the starting vertex

at each stage of the algorithm:

• the original graph minus the darkened (already used) edges = reduced graph
• important rule: never cross a bridge of the reduced graph unless there is no other choice
• why must we observe that rule?

• the same algorithm works for Euler paths
• before starting, use Euler’s theorems to check that the graph has an Euler path and/or circuit to find!
• when you do this on paper, you can erase each edge as you traverse it
• this will make the reduced graph visible, and its bridges apparent

Step in recipe	Marked graph	Reduced graph
Pick any vertex (e.g. F)
Travel from F to C  (arbitrary choice)
Travel from C to D  (arbitrary)
Travel from D to A  (arbitrary)
Travel from A to C  (can't go to B: that edge is a bridge of the reduced graph, and there are two other choices, we chose one of them)

The rest of the trip is obvious, and the complete Euler circuit is:

(F, C, D, A, C, E, A, B, D, F)