We know that 4-connected planar graphs are Hamiltonian(by the known Tutte Theorem). Additionally, Thomas and Yu [1] proved that removing two vertices from a 4-connected planar graph still preserves Hamiltonicity. My question is: is a 4-connected planar graph still Hamiltonian after removing an edge? I'm not sure if there are any existing results on this.
In a more formal tone, my question would be phrased as:
Question: Let $G$ be a $4$-connected planar graph. Let $G'$ be a graph obtained from $G$ by deleting an edge in $G$. Then is $G'$ Hamiltonian?
[1] R. Thomas and X. Yu, 4-connected projective-planar graphs are Hamiltonian, J. Combin. Theory Ser. B, 62 (1994), pp. 114--132.
