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Jul 22, 2020 at 21:46 comment added gmath Just a thought: if you think of the dual, I think you are looking at number of ways of connecting the edges of 6 vertices of degree 6, so that the final graph is connected. Since you have numbered the sides, you would also need to number the edges. Maybe looking at the configuration model (en.wikipedia.org/wiki/Configuration_model) is useful.
Jul 21, 2020 at 23:06 comment added Gerry Myerson oeis.org/A000228 may give you a start.
Jul 21, 2020 at 23:03 comment added Gerry Myerson Astronomical numbers are quite small, by mathematical standards.
Jul 21, 2020 at 12:01 history edited Martin Sleziak CC BY-SA 4.0
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Jul 21, 2020 at 11:59 history edited YCor CC BY-SA 4.0
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Jul 21, 2020 at 11:15 history asked Matt Roberts CC BY-SA 4.0