![](/files/happy5.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ r > 2 $](/files/tex/97a6c1471f5d46c5d21bf3e382139df59f21bd80.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
(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.