# r-regular graphs are not uniquely hamiltonian.

**Conjecture**If is a finite -regular graph, where , then is not uniquely hamiltonian.

(Reproduced from [M].)

A graph is said to be *uniquely hamiltonian* if it contains precisely one Hamiltonian cycle.

This conjecture has been proved for all odd values of [T] and for all even values of [H]. By Petersen's theorem, it would suffice to prove it for .

## Bibliography

[H] P. Haxell, Oberwolfach reports, 2006.

[M] Bojan Mohar, Problem of the Month

*[S] John Sheehan: The multiplicity of Hamiltonian circuits in a graph. Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pp. 477-480. Academia, Prague, 1975, MathSciNet

[T] A.G. Thomason, Hamiltonian cycles and uniquely edge colourable graphs. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). Ann. Discrete Math. 3 (1978), Exp. No. 13, 3 pp.

* indicates original appearance(s) of problem.

### The construction in that

The construction in that paper has parallel edges, so it is not a counter example. As far as I am aware, Sheehan's conjecture is still open.

### The conjecture is only for

The conjecture is only for simple graphs. The paper you mention gives counterexamples that have multiple edges.

## Is this question still open?

I think, the autor answers the question in this article: Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs (https://doi.org/10.1002/jgt.3190180503). (And the conjecture is fals for all even values.) Am I wrong?