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We fully disprove a conjecture of Haythorpe on the minimum number ofhamiltonian cycles in regular hamiltonian graphs, thereby extending a result ofZamfirescu, as well as correct and complement Haythorpe's computationalenumerative results from [Experim. Math. 27 (2018) 426-430]. Thereafter, we usethe Lov\'asz Local Lemma to extend Thomassen's independent dominating setmethod. Regarding the limitations of this method, we answer a question ofHaxell, Seamone, and Verstraete, and settle the first open case of a problem ofThomassen. Motivated by an observation of Aldred and Thomassen, we prove thatfor every $\kappa \in \{ 2, 3 \}$ and any positive integer $k$, there areinfinitely many non-regular graphs of connectivity $\kappa$ containing exactlyone hamiltonian cycle and in which every vertex has degree $3$ or $2k$.

Few hamiltonian cycles in graphs with one or two vertex degrees