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  • Which degree sequences are planar-graphical, that means for which degree sequences $$d_1,...,d_n$$ $$d_1\le...\le d_n$$ exists a PLANAR graph that has this degree sequence ?

    I found some links in the internet, but I did not find a concrete classification. I know Euler's polyeder formula and that at least one vertex must have degree less than $6$ and some similar restrictions, but I would like to have some more powerful conditions.

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There is no complete classification of the degree sequences of planar graphs as of today. However, I do have an example for you of an infinite number of similar degree sequences that always contain a planar graph:

$ (N,N,N,N,4,4,4,4,...) $, where $ N > 4 $ and the number of $ 4's $ in the sequence is equal to $ (N-2)^2 $.

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  • $\begingroup$ This seems like a very trivial result in this direction. Maybe a bit more interesting is that an $r$-regular planar graph on $n$ vertices always exists provided (1) $rn$ is even, the necessary condition for regular graphs (2) $\frac{rn}{2} \le 3n-6$, Euler's condition for planar graphs, and (3) $(r,n)$ is not $(4,7)$ or $(5,14)$. (source, I think) $\endgroup$ – Misha Lavrov Jul 18 '18 at 21:58
  • $\begingroup$ @MishaLavrov What makes it so trivial? $\endgroup$ – Ultradark Jul 18 '18 at 22:01
  • $\begingroup$ The fact that it's only about one very specific family of degree sequences. $\endgroup$ – Misha Lavrov Jul 18 '18 at 22:31
  • $\begingroup$ Just because it is specific does not mean it's trivial. $\endgroup$ – Ultradark Aug 8 '18 at 20:52

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