Geodesic
A New Proof that 4-Connected Planar Graphs are Hamiltonian-Connected
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 555-564.

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We prove a theorem guaranteeing special paths of faces in 2-connected plane graphs. As a corollary, we obtain a new proof of Thomassen’s theorem that every 4-connected planar graph is Hamiltonian-connected.
Keywords: 4-connected planar graph, Hamiltonian-connected, Tutte-path 
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Lu, Xiaoyun; West, Douglas B. A New Proof that 4-Connected Planar Graphs are Hamiltonian-Connected. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 555-564. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a3/

[1] C. Thomassen, A theorem on paths in planar graphs, J. Graph Theory 7 (1983) 169-176. doi:10.1002/jgt.3190070205

[2] W.T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956) 99-116. doi:10.1090/S0002-9947-1956-0081471-8

[3] H. Whitney, A theorem on graphs, Ann. of Math. 32 (1931) 378-390. doi:10.2307/1968197