1. We learned that if a graph has exactly two vertices with odd valences, then an Euler circuit does not exist—but an Euler path does. It is also possible to produce an Euler circuit through the process of eulerization, by duplicating certain edges of the graph. But how many duplications are necessary to obtain an Euler circuit? Investigate this problem and more general related topics using the Eulerizing a Graph applet.

2. We know that if all the vertices have even valence, then an Euler circuit exists. Try your hand at finding such circuits in the Euler Circuit applet.