A059103 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.

1, 1, 2, 5, 13, 51, 222, 1313, 9639

Offset: 1

Author

David S. Newman, Feb 13 2001

Keywords

• hard
• more
• nonn

Comments

This counting problem is related to finding the chromatic number of the plane, X(R^2).

Examples

			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.
		

References

• 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.

Links

• Aidan Globus and Hans Parshall, Small unit-distance graphs in the plane, arXiv:1905.07829 [math.CO], 2019.
• Matthew McAndrews, Simple Connected Units Distance Graphs Through 6 Vertices
• Eric Weisstein's World of Mathematics, Connected Graph
• Eric Weisstein's World of Mathematics, Unit-Distance Graph

Crossrefs

Cf. A350507 (not necessarily connected unit-distance graphs).

Cf. A303792 (connected matchstick graphs).

Cf. A308349 (minimal unit-distance forbidden graphs).

Extensions

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

a(7) from Hans Parshall, May 03 2018

a(8)-a(9) from Hans Parshall, May 21 2019