Tesselate a torus with finitely many simply connected polygons. Do not allow four or more of them to meet at a point. In counting the edges, don't count a "straight line" as just one edge if it's the boundary between polygons A and B until you reach a point after which it's the boundary between A and C; at that point one edge ends and the next starts. (You might say "straight line" means a geodesic, but we maybe don't need to be so sophisticated: just say there's a $180^\circ$ angle there, not $179^\circ$, etc.)
Then: The average number of edges of the tesselating polygons is exactly 6.
Proof: $V-E+F=0$, then massage.
The question: Is the statement after "then" in citable literature somewhere?
Later comment: It may seem odd to include the note about counting edges in a graph, since it's the only way anyone would count them, but when one thinks of counting edges of a polygon, it may seem odd to think of one of the four sides of a rectangle as two edges rather than one.
This doesn't exactly answer your reference request, but here is an example where the polygons are all hexagons (not just on average.) http://en.wikipedia.org/wiki/Szilassi_polyhedron