#### Answer

False

#### Work Step by Step

Not every complete graph that has a Hamilton circuit has at least one Euler circuit. There are complete graphs with all odd vertices which have a Hamilton circuit. However, since the graph has odd vertices, the graph can not have an Euler circuit. A graph can only have an Euler circuit if all the vertices are even.
For example, let's consider a complete graph with four vertices. Then the graph has (4-1)! Hamilton circuits (which is 6 Hamilton circuits). However, since all the vertices are odd, the graph has no Euler circuit.