Graph classes where Hamiltonian Cycle and Hamiltonian Path problems have different complexity While searching The information System on Graph Classes and their Inclusions, I found several graph classes for which  the Hamiltonian Cycle problem is NP-complete while the complexity of Hamiltonian Path problems is NOT known. Some of those classes are bipartite maximum degree 3 graphs, maximum degree 3 grid graphs, and 2-connected cubic planar graphs. Also this phenomena applies to circle graphs and triangular grid graphs.


  Is there an update to the complexity of Hamiltonian path problem on those classes? Is there an explanation for this phenomena?


EDIT: I found in the graph classes database a weird case of solid grid graphs where Hamiltonian cycle problem is in $P$ while Hamiltonian path problem is of unknown complexity.
The Hamiltonian path problem on grid graphs with maximum degree 3 is NP-complete. The proof is in C. H. Papadimitriou and U. V. Vazirani, On two geometric problems related to the travelling salesman problem, Journal of Algorithms, Volume 5, Issue 2, June 1984, Pages 231–246 (Theorem 2)