# Hamiltonicity of Cayley graphs

**Question**Is every Cayley graph Hamiltonian?

This problem seems to have been first considered by Rapaport-Strasser [R]. Lovasz [L] conjectured more generally that every vertex-transitive graph is Hamiltonian. For a survey of results up to 1996, see [CG]. Although many specific Cayley graphs have been shown to be Hamiltonian, there are few general results. One exception is the theorem by Pak and Radoičić [PR] that every finite group with at least three elements has a generating set of size , such that the corresponding Cayley graph is Hamiltonian.

## Bibliography

[CG] S. J. Curran, J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey, Discrete Math. 156 (1996), 1–18.

[L] L. Lovasz, Problem 11, in “Combinatorial structures and their applications,” University of Calgary, Calgary, Alberta, Canada (1970), Gordon and Breach, New York.

[PR] I. Pak and R. Radoičić, Hamiltonian paths in Cayley graphs, preprint.

*[R] E. Rapaport-Strasser, Cayley color groups and Hamilton lines, Scripta Math. 24 (1959), 51–58.

* indicates original appearance(s) of problem.

### [PR] paper

First, link to PR paper has moved with Pak homepage. Second, it is now published in *Discrete Math.*

### Is this at all relevant?

Not sure but wondering if this link is at all relevant to the problem:

http://books.google.com/books?id=aSyXqtfOuU4C&pg=PA49&dq=Cayley+graph&num=100&client=firefox-a#PPA49,M1

Thanks,

- Farley

### graphs vs. digraphs

The book you link to has examples of directed Cayley graphs with no (directed) Hamiltonian cycle. The existence of such graphs is interesting and relevant, but does not give a counterexample to the problem stated here.

## Hamiltonicity of dence Cayley graphs

Christofides, Hladky, and Mathe (http://arxiv.org/abs/1008.2193) proved using the Regularity Method the case when the vertex-transitive graph is sufficiently dense.