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%I #58 Jan 03 2022 11:08:31
%S 1,1,2,5,13,51,222,1313,9639
%N Number of simple, connected, unit-distance graphs on n points realizable in the plane with straight edges all of the same length; lines are permitted to cross.
%C This counting problem is related to finding the chromatic number of the plane, X(R^2).
%D K. B. Chilakamarri and C. R. Mahoney, Maximal and minimal forbidden unit-distance graphs in the plane, Bulletin of the ICA, 13 (1995), 35-43.
%H Aidan Globus and Hans Parshall, <a href="https://arxiv.org/abs/1905.07829">Small unit-distance graphs in the plane</a>, arXiv:1905.07829 [math.CO], 2019.
%H Matthew McAndrews, <a href="/A059103/a059103.pdf">Simple Connected Units Distance Graphs Through 6 Vertices</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Unit-DistanceGraph.html">Unit-Distance Graph</a>
%e a(4)=5 because the complete graph on 4 points cannot be realized in the plane with all edges of equal length. All the other connected graphs with 4 points can be realized.
%Y Cf. A350507 (not necessarily connected unit-distance graphs).
%Y Cf. A303792 (connected matchstick graphs).
%Y Cf. A308349 (minimal unit-distance forbidden graphs).
%K hard,more,nonn
%O 1,3
%A _David S. Newman_, Feb 13 2001
%E a(6) has been updated to reflect the fact that it has recently been proved to be 51 rather than 50. - _Matthew McAndrews_, Feb 21 2016
%E a(7) from _Hans Parshall_, May 03 2018
%E a(8)-a(9) from _Hans Parshall_, May 21 2019
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