A328061 Number of 4-chromatic Laman graphs on n vertices.

1, 8, 102, 1601, 29811, 636686, 15298955, 407748141, 11932078866

Offset: 7

Author

Vsevolod Voronov, Oct 03 2019

Keywords

• nonn
• more

Comments

All the Laman graphs (in other words, minimally rigid graphs) can be constructed by the inductive Henneberg construction, i.e., a sequence of Henneberg steps starting from K_2. A new vertex added by a Henneberg move is connected with two or three of the previously existing vertices. Hence, the chromatic number of a Laman graph can be 2, 3 or 4. One can hypothesize that the set of 3-chromatic Laman graphs is the largest and that bipartite graphs are relatively rare. The simplest example of a 4-chromatic Laman graph is the Moser spindle.

Links

• L. Henneberg, Die graphische Statik der starren Systeme, Leipzig, 1911.
• Christoph Koutschan, Mathematica program for generating a list of non-isomorphic Laman graphs on n vertices.
• G. Laman, On Graphs and Rigidity of Plane Skeletal Structures, J. Engineering Mathematics, Vol. 4, No. 4, 1970, pp. 331-340; alternative link.
• Martin Larsson, Nauty Laman plugin
• A. Nixon, E. Ross, One brick at a time: a survey of inductive constructions in rigidity theory, arXiv:1203.6623 [math.MG], 2012-2013.
• Vsevolod Voronov, Anna Neopryatnaya, and Eugene Dergachev, Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres, arXiv:2106.11824 [math.CO], 2021.
• Eric Weisstein's World of Mathematics, Moser spindle is a 4-chromatic Laman graph.
• Wikipedia, Laman graph

Crossrefs

Cf. A227117, A273468, A328060.

Programs

Mathematica

Table[Length[
  Select[LamanGraphs[n],
   IGChromaticNumber[AdjacencyGraph[G2Mat[#]]] == 4 &]], {n, 7, 9}]
(*  using the program by Christoph Koutschan for generating Laman graphs, see A227117, and IGraph/M interface by Szabolcs Horvát *)

nauty

gensparseg $n -K2 | countg --N # With Laman plugin; see link.

Extensions

a(13)-a(15) added by Georg Grasegger, Sep 09 2024