Every 4-connected optimal 2-planar graph is Hamiltonian-connected, with the 4-connectedness condition being sharp via infinitely many 3-connected counterexamples that are non-Hamiltonian.
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A note on optimal 2-planar graphs
Every 4-connected optimal 2-planar graph is Hamiltonian-connected, with the 4-connectedness condition being sharp via infinitely many 3-connected counterexamples that are non-Hamiltonian.