A graph is uniquely pancyclic if it contains exactly
one cycle for each possible length (from 3 to the order
of the graph).
There are seven known graphs that are uniquely pancyclic.
One of order 3, one of order 5, and the five graphs
shown below (click on any of the thumbnails for a larger
picture).
Finding all such graphs is unsolved problem number 10
on the list contained in **Graph Theory with Applications**
by Bondy and Murty.

To get an idea what is going on, I tried to construct graphs that
contain exactly two cycles of each possible length (call these
graphs **2-pancyclic**). Some of these are shown below.

n = 11 | n = 13 | n = 17 | n = 19 |

Images created with PerlMagick and Gwiz.