The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths in L-alphabet, C-alphabet, F-alphabet, and E-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.
Fatemeh Keshavarz-Kohjerdi. Alireza Bagheri. "Hamiltonian Paths in Some Classes of Grid Graphs." J. Appl. Math. 2012 1 - 17, 2012. https://doi.org/10.1155/2012/475087