$G$ is a simple undirected, regular, and planar graph with 9 vertices. Prove or disprove that $\bar G$ must have a Hamiltonian cycle.

Question

Original question: $G$ is a simple undirected planar graph with 9 vertices. Suppose that every vertex in G has the same degree (regular). Prove or disprove that the complementary graph $\bar G$ must have a Hamiltonian cycle.

This is part of a past paper for a discrete math course, but the answers were not provided. I have tried to do it for a while now, but I don't even have any idea how to start this proof. Could someone please point me in the right direction, or even better, show me how to do this?

Answer 1

Planar graph on $n$ vertices has at most $3n - 6$ edges if $n \ge 3$ (this follows from Euler's formula). Since $G$ is regular graph on $9$ vertices we have $\deg G \le \lfloor \frac{2\cdot 21}{9}\rfloor = 4$. Then $\deg \overline{G} \ge 4$. If $\deg \overline{G} \ge 5$ then it is Hamiltonian by Dirac's theorem. The remaning case is $\deg \overline{G} = 4$.

Futher my solution becomes rather sophisticated in spite of OEIS tells that are at most 16 possible graphs $\overline{G}$ left to consider. So I just point that all of them are Hamiltonian.