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



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> Are there any 4-regular planar graphs without 2-cycles + 3-cycles?
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Could someone draw an example if there is one? I couldn't find any paper that answers this question.



Even regular planar graphs without 2-cycles
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 $E+2=V+F$.


