#### Answer

The path F,G,E,C,B,A,D,F is a Hamilton circuit that begins at F and ends with the pair of vertices D,F.

#### Work Step by Step

A Hamilton path is a sequence of adjacent vertices that visits each vertex in the graph exactly once. A Hamilton circuit is a Hamilton path that begins and ends at the same vertex.
Let's start at vertex F. Note that moving to vertex G must be the first step, otherwise the path will get "stuck" at vertex G if we try to move there from vertex E.
The path must travel from vertex F to vertex G, then to vertex E, then to vertex C, then to vertex B, then to vertex A, then to vertex D, and then finally to vertex F.
This path is F,G,E,C,B,A,D,F. Since this path visits every vertex in the graph exactly once (except for vertex F which is the starting and ending vertex), this path is a Hamilton path. Since this Hamilton path starts and ends at the same vertex, this path is a Hamilton circuit.
This is the only Hamilton circuit in the graph that begins at vertex F and ends with the pair of vertices D,F.