Pancyclic Graph

A pancyclic graph is a simple unlabeled graph on vertices that contains cycles of all lengths, 3, 4, ..., . Since a pancyclic graph must contain a graph cycle of length , pancyclic graphs are of necessity Hamiltonian.

The numbers of pancyclic graphs on , 2, ... vertices are 0, 0, 1, 2, 7, 43, 372, 6132, 176797, 9302828, ... (OEIS A286684), the first few of which are illustrated above.

Classes of graphs which are pancyclic include:

1. antiprism graphs,

2. Chang graphs,

3. dipyramidal graphs,

4. Johnson graphs,

5. Mathon graphs,

6. Paulus graphs,

7. Sierpiński gasket graphs,

8. sun graphs,

9. tetrahedral graphs, and

10. wheel graphs.

Pancyclic graphs that have exactly one graph cycle of each length are very rare and are known as uniquely pancyclic graphs.

A combination of graph toughness and a sufficiently large edge count forces pancyclicity. If is an integer and is a t connected graph of graph order and edge count , then

implies that is pancyclic (Jia et al. 2026).