# Almost all non-Hamiltonian 3-regular graphs are 1-connected

**Conjecture**Denote by the number of non-Hamiltonian 3-regular graphs of size , and similarly denote by the number of non-Hamiltonian 3-regular 1-connected graphs of size .

Is it true that ?

A stronger version of this conjecture asks whether it is also the case that for all .

Experimental data was given by Filar et al [FHN] demonstrating that the strong conjecture is satisfied for all , and with sampled data provided for and . No further results have been forthcoming.

The experimental data can be viewed at http://dx.doi.org/10.7151/dmgt.1485

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## Bibliography

[FHN] Jerzy A Filar, Giang T Nguyen, Michael Haythorpe, "A conjecture on the prevalence of cubic bridge graphs", Discussiones Mathematicae Graph Theory 30(1):175--179 (2010).

* indicates original appearance(s) of problem.