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

1,3

Comments

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

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

Table of n, a(n) for n=1..9.

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

Example

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.

Crossrefs

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

Cf. A303792 (connected matchstick graphs).

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

Sequence in context: A098716 A082938 A303792 * A365709 A260709 A112836

Adjacent sequences: A059100 A059101 A059102 * A059104 A059105 A059106

Keyword

hard,more,nonn

Author

David S. Newman, Feb 13 2001

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

Status

approved