I have a homework for my class to Combinatorics and Graphs which I'm not sure how to finish.
The task: Let G be a simple graph on 14 vertices, with 4 vertices having degree 5 and 10 vertices having degree 7. Prove that G is Hamiltonian.
We've went through Dirac's and Ore's theorems, but I'm not sure how to use them (at least at the start of the proof), and also Chvátal-Erdős theorem, which seems more appropriate to begin with.
Where did I come so far: We know that the number of edges in this graph is 45 (that is, 4*5 + 10*7 divided by two, as we count every edge twice through degrees), and the $K_1$$_4$ has 91 edges (($14,2$) - combination number), so we still could somehow add 91-45 = 46 edges. Now, from one variation of C-E theorem, I know that:
G is Hamiltonian $\iff G=(V,E_+)$ is Hamiltonian, where E+ is E plus edge between two not adjacent vertices whose sum of degrees is larger or equal to n. With knowing that I can add 46 edges, I can get to the point when I know that two 7-deg vertices are not connected - there's simple not enough vertices between them. So I can add an edge, and what I wanted to prove is that I can add the edges in a way that there will be then two at-least-9-deg vertex for each 5-deg vertex, and so I'd be able to connect them in a same way as I did with 7-deg vertices, up to the point when every 5-deg vertex would have degree of 7, after which point I could use Dirac's theorem to prove that G is Hamiltonian. However, this doesn't seem like the good way, or at least I'm not entirely convinced about it.
Could anybody please comment wether this is the good way to go, or better yet, point me in the right Dirac'tion for the proof? Thanks!!