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Parts Graphs


PartsGraphs

The Parts graphs are a set of unit-distance graphs with chromatic number five derived by Jaan Parts in 2019-2020 (Parts 2020a). They provide some of the smallest known examples that establish the solution to the Hadwiger-Nelson problem (i.e., the chromatic number of the plane) as 5, 6, or 7.

The Parts graphs are summarized in the following table and illustrated above.

vertex countedge countdiscovery date
5532840Jul. 4, 2019
5292630Jul. 4, 2019
5252605Jul. 16, 2019
5102508Aug. 3, 2019
5102502prior to Mar. 7, 2020
5092442prior to Mar. 7, 2020
PartsGraphs2

Additional graphs on 16, 31, and 199 nodes are also associated with Parts (2020b). The 31-node graph gives a small 6-chromatic graphs with exactly two edge lengths (1 and the golden ratio phi). It can be obtained by combining one copy of the 16-vertex graph with another obtained by rotating about one of the first copy's vertices. (Note that note that both the 16- and 31-node graphs are edge-edge and edge-vertex degenerate.)

The Parts graphs are implemented in the Wolfram Language as GraphData["PartsGraph509"] etc.


See also

de Grey Graphs, Hadwiger-Nelson Problem, Heule Graphs, Mixon Graphs, Unit-Distance Graph

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References

--. "Hadwiger-Nelson Problem." https://asone.ai/polymath/index.php?title=Hadwiger-Nelson_problem.de Grey, A. D. N. J. "The Chromatic Number of the Plane Is at Least 5." Geombinatorics 28, No. 1, 18-31, 2018.Heule, M. J. H. "Computing Small Unit-Distance Graphs with Chromatic Number 5." Geombinatorics 28, 32-50, 2018.Parts, J. Polymath16 comments. https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713, https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-deadline/#comment-23814, and https://dustingmixon.wordpress.com/2019/03/23/polymath16-twelfth-thread-year-in-review-and-future-plans/#comment-23713.Parts, J. "Graph Minimization, Focusing on the Example of 5-Chromatic Unit-Distance Graphs in the Plane." Geombinatorics 29, No. 4, 137-166, 2020a.Parts, J. "A Small 6-Chromatic Two-Distance Graph in the Plane." 23 Oct 2020b. https://arxiv.org/abs/2010.12656.Soifer, A. "Jaan Parts' Current World Record." Ch. 56 in The New Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators, 2nd ed. New York: Springer, pp. 701-708, 2024.

Cite this as:

Weisstein, Eric W. "Parts Graphs." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PartsGraphs.html

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