# Uniquely hamiltonian graphs with minimum degree 4

A graph is uniquely hamiltonian if it has exactly one Hamilton cycle.

As every edge in a cubic graph lies in an even number of Hamilton cycles, a cubic graph cannot be uniquely hamiltonian, and a famous conjecture, called Sheehan's conjecture asserts that a 4-regular graph can also not be uniquely hamiltonian.

Apparently, however, there are uniquely hamiltonian graphs with minimum degree equal to four - the latest edition of Bondy & Murty's Graph Theory even gives a reference to a paper by H. Fleischner entitled "Uniquely hamiltonian graphs of minimum degree 4", To Appear, Journal of Graph Theory and dates it at 2007.

But I cannot find this paper on MathSciNet and nor can I find any paper that appears to contains these graphs. Googling reveals a few references to this work, including a 2003 conference/workshop where the abstract claims that there are some uniquely hamiltonian Eulerian graphs with minimum degree four.

Does anyone know what these graphs are? Or where this paper is? Or anything at all?

(I have emailed Fleischner but had no reply yet, though I may still get one as the email was recent.)

EDIT: Due to some bizarre synchronicity, I received a reply from Herbert Fleischner literally 5 seconds after I finished writing this question; the paper exists and has been accepted by JGT but final revisions have not been made.

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If $G$ is a finite $r$-regular graph, where $r > 2$, then $G$ is not uniquely Hamiltonian.