#### Answer

All the vertices in the connected graph are even vertices. Therefore, according to Euler's theorem, there is at least one Euler circuit in the graph.
The mail carrier will be able to travel through every edge on the graph exactly once without retracing any edge. Also, the mail carrier will be able to return to vertex B.

#### Work Step by Step

All the vertices in the connected graph are even vertices. Therefore, according to Euler's theorem, there is at least one Euler circuit in the graph.
An Euler path is a path that travels through every edge of a graph exactly once. An Euler circuit is an Euler path that begins and ends at the same vertex.
Following an Euler circuit, the mail carrier will be able to travel through every edge on the graph exactly once without retracing any edge. Also, since an Euler circuit begins and ends at the same vertex, the mail carrier is able to return to vertex B.