The Harborth graph is the smallest known 4-regularmatchstick graph. It is therefore both planar
and unit-distance. It has 104 edges
and 52 vertices. This graph was named after its discoverer
H. Harborth, who first presented it to a general public in 1986 (Harborth 1994,
Peterson 1996, Gerbracht 2006, Winkler et al. 2017).
Bode, J.-P.; Harborth, H.; and Thürmann, C. "Minimum Regular Rectilinear Plane Graph Drawings with Fixed Numbers of Edge Lengths."
Congr. Numer.169, 193-198, 2004.Gerbracht, E. H.-A.
"Minimal Polynomials for the Coordinates of the Harborth Graph." 5 Oct
2006. https://arxiv.org/abs/math/0609360.Harborth,
H. "Match Sticks in the Plane." In The Lighter Side of Mathematics.
Proceedings of the Eugéne Strens Memorial Conference of Recreational Mathematics
& its History. Calgary, Canada, July 27-August 2, 1986 (Ed. R. K. Guy
and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., pp. 281-288,
1994.Harborth, H. and Kemnitz, A. "Integral Representations of
Graphs." In Contemporary Methods in Graph Theory (Ed. R. Bodendiek).
Mannheim, Germany: B.I.-Wissenschaftsverlag, pp. 359-367, 1990.Hartsfield,
N. and Ringel, G. Pearls
in Graph Theory: A Comprehensive Introduction. San Diego, CA: Academic Press,
1990.House of Graphs. "Harborth Graph." https://houseofgraphs.org/graphs/1144.Kurz,
S. "No Finite 5-Regular Matchstick Graph Exists." 8 Jan 2014. https://arxiv.org/abs/1401.1793.Kurz,
S. and Pinchasi, R. "Regular Matchstick Graphs." Amer. Math. Monthly118,
264-267, 2011.Pegg, E. Jr. "MathPuzzle: Material added 8 Jan
06 (Happy New Year)." Jan. 8, 2006. https://www.mathpuzzle.com/26Feb2006.html.Peterson,
I. "Mathland: Matchsticks in the Summer." August 1996. https://web.archive.org/web/20080322001146/http://www.sciencenews.org/pages/sn_arch/8_10_96/mathland.htm.Winkler,
M.; Dinkelacker, P.; and Vogel, S. "New Minimal -Regular Matchstick Graphs." Geombinatorics27,
26-44, Jul. 2017.
-Regular Matchstick Graphs." Geombinatorics 27,
26-44, Jul. 2017.