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Number of 4-chromatic Laman graphs on n vertices.
2

%I #42 Sep 09 2024 06:26:42

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

%N Number of 4-chromatic Laman graphs on n vertices.

%C 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.

%H L. Henneberg, <a href="https://archive.org/details/diegraphischest00henngoog">Die graphische Statik der starren Systeme</a>, Leipzig, 1911.

%H Christoph Koutschan, <a href="https://oeis.org/A227117/a227117_2.txt">Mathematica program</a> for generating a list of non-isomorphic Laman graphs on n vertices.

%H G. Laman, <a href="https://doi.org/10.1007/BF01534980">On Graphs and Rigidity of Plane Skeletal Structures</a>, J. Engineering Mathematics, Vol. 4, No. 4, 1970, pp. 331-340; <a href="https://platformwiskunde.nl/wp-content/uploads/2021/02/ref_jenma.pdf">alternative link</a>.

%H Martin Larsson, <a href="https://github.com/martinkjlarsson/nauty-laman-plugin">Nauty Laman plugin</a>

%H A. Nixon, E. Ross, <a href="https://arxiv.org/abs/1203.6623">One brick at a time: a survey of inductive constructions in rigidity theory</a>, arXiv:1203.6623 [math.MG], 2012-2013.

%H Vsevolod Voronov, Anna Neopryatnaya, and Eugene Dergachev, <a href="https://arxiv.org/abs/2106.11824">Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres</a>, arXiv:2106.11824 [math.CO], 2021.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoserSpindle.html">Moser spindle</a> is a 4-chromatic Laman graph.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laman_graph">Laman graph</a>

%t Table[Length[

%t Select[LamanGraphs[n],

%t IGChromaticNumber[AdjacencyGraph[G2Mat[#]]] == 4 &]], {n, 7, 9}]

%t (* using the program by Christoph Koutschan for generating Laman graphs, see A227117, and IGraph/M interface by Szabolcs Horvát *)

%o (nauty with Laman plugin) gensparseg $n -K2 | countg --N # see link

%Y Cf. A227117, A273468, A328060.

%K nonn,more

%O 7,2

%A _Vsevolod Voronov_, Oct 03 2019

%E a(13)-a(15) added by _Georg Grasegger_, Sep 09 2024