Proof

By hypothesis, $G$ is a simple, connected, and planar graph with a dual graph $G*$ which is Hamiltonian.
Since $G^*$ is Hamiltonian, by the characterization of planar hamiltonian graphs, the minimal tree decomposability of $G$ equals $2.$
By the definition of tree decomposability, $G$ can be therefore decomposed into exactly $2$ trees.
Each of these two trees can be colored with two colors, since any tree can be colored with two colors.
Therefore, we can choose two colors for the first tree and some other two colors for the second tree.
Thus, its chromatic number obeys the inequality $\chi(G)\le 2\cdot 2=4.$
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