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|
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