(related to Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph)

- 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 $\chi(G)\le 2\cdot 2=4.$∎

**Piotrowski, Andreas**: Own Research, 2014