TOPICS

# Pancyclic Graph

A simple unlabeled graph on vertices is called pancyclic if it contains cycles of all lengths, 3, 4, ..., . Since a pancyclic graph must contain a 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:

2. Chang graphs,

8. sun graphs,

9. tetrahedral graphs, and

10. wheel graphs.

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

Graph Cycle, Hamiltonian Graph, Uniquely Pancyclic Graph

## Explore with Wolfram|Alpha

More things to try:

## References

George, J. C.; Khodkar, A.; and Wallis, W. D. Pancyclic and Bipancyclic Graphs. Cham, Switzerland: Springer, 2016.Sloane, N. J. A. Sequence A286684 in "The On-Line Encyclopedia of Integer Sequences."

Pancyclic Graph

## Cite this as:

Weisstein, Eric W. "Pancyclic Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PancyclicGraph.html