And real women (like my wife, who is a lurker on this forum) design origami modules and link them together to investigate similar things. I will try to think more about this over the weekend; maybe we can get enough terms to search OEIS. Thank you for an enjoyable problem, Warren. On Fri, Jan 31, 2014 at 6:47 PM, Warren D Smith <warren.wds@gmail.com>wrote:
I have no confidence at all that every squarefree graph is realizable.
In fact, I can actually disprove this, except my disproof currently cheats slightly. Namely, the two papers I cited before show that maximal squarefree graph has order V^(3/2) edges, whereas unit-distance graph has O(V^(4/3)) edges, hence not every squarefree graph is realizable as a unit distance graph, QED. The reasons this "cheated" is I never verified the Szekely paper's V^(4/3) bound works on spheres. But aside from that, this seems solid.
So anyhow, the squarefree graph thing merely yields an upper bound on f(N).
But you can for any particular N work out f(N) by machine by considering every possible N-vertex graph and deciding if it is realizable. It ain't an easy job, but it is doable in principle. In fact you can probably get up to maybe N=10 just manually. We don't need no stinking computers, real men cut triangular thingummies with their bowie knives and move them around on spheres to solve such things.
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