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graphs that can be embedded into the plane, i.e. that can be drawn without crossings between the lines representing edges.
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There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....?
There is a 3-connected 5-regular simple $n$-vertex planar graph if and only if $n=12$ or $n \ge 16$ is even. See Recursive generation of 5-regular graphs by Mahdieh Hasheminezhad, Brendan D. McKay, T …