A graph $G$ with $n$ vertices is called pancyclic if it contains cycles of length $m$, for every value of $m$ such that $3 \le m \le n$. The question as stated originates from the article [1] by Malkevitch et al., who posed three problems. I am particularly interested in the first problem, but I am not sure if it has already been solved by someone, as it has been over 50 years.
Problem 1 [1] Do there exist 5-valent planar, 3-connected (4 or 5 connected?) hamiltonian graphs which are not pancyclic?
Perhaps this question is just a routine exercise? My intuition tells me it exists.
[1] J. Malkevitch, On lengths of cycles in planar graphs. Recent Trends in Graph Theory. Springer-Verlag, Berlin (1971) 191-195.