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Related to another question I asked, some questions came up, the most important is the following:

Are there any 4-regular planar graphs without 2-cycles + 3-cycles?

Could someone draw an example if there is one? I couldn't find any paper that answers this question.

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In a connected planar 4-regular graph without cycles of length less than 4 we have $E=2V$ and $2E\geqslant 4F$ (since every face has at least 4 edges and every edge belongs to at most 2 faces). Thus $2E\geqslant 2V+2F$, contradiction to Euler formula $E+2=V+F$.

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For a plane 4-valent connected graph without loops twice the number of 2-gons plus the number of 3-gons must be at least 8, a result that follows from Euler’s formula.

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There is an infinite example: the infinite grid-graph!

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  • $\begingroup$ I wasn't precise in my question. I was interested in S^3 , a rather finite one :D $\endgroup$
    – Kregnach
    Commented Aug 11, 2023 at 20:31
  • $\begingroup$ Ah yes. I was going to argue that this is still interesting. But you could even do stupider things like the infinite 4-regular tree, so this isn't unique or anything. $\endgroup$
    – Zach Hunter
    Commented Aug 12, 2023 at 1:44
  • $\begingroup$ maybe you want to check out my previous question on the platform, that is related to this one (still no answer) > "Three-dimensional triangulations with fixed number of vertices" $\endgroup$
    – Kregnach
    Commented Aug 12, 2023 at 21:01

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