Prove that G is Hamiltonian

Question

Given $G$ a graph with degrees:$6,6,4,4,4,k,k$ on $7$ vertices and $10$ regions
(and by Euler $n-f+r=2$ I found that $k$=3) prove $G$ is contains a Hamiltonian cycle

I did find a visual cycle on the actual graph, where in the solutions (by a student) he proved that vertex 3 doesn't have to be a neighbor with the other vertex degree 3 and thus by the theorem the graph is hamiltonian

What will be a better explanation in a discrete math course

Answer 1

Here it is... a Hamiltonian cycle in your graph:

Answer 2

If the sum of every pair of vertices is at least n than your graph has a Hamiltonian cycle. Also if you can prove that this graph has a cyclic graph C7 as a subgraph than it is also Hamiltonian.