The Grabarchuk graph is the Euclidean distance graph obtained from the vertices of a 
grid graph when
two vertices are considered adjacent
when the Euclidean distance between them is
3. Since 
has two representations as three squares,
 |  |  |
(1)
|
 |  |  |
(2)
|
edges correspond to points in the grid that are either distance 3 apart along a line parallel to a Cartesian axis, or else distances 1, 2, and 2 apart along the Cartesian axes.
The Grabarchuk graph is illustrated above in an order-4 LCF notation, a bilaterally symmetric order-1 LCF notation, a bipartite graph drawing, and two voltage graph drawings.
The resulting graph is a sextic graph (Knuth 2025, p. 60).
The graph is named after Serhiy Grabarchuk, who communicated it to D. Knuth in 2018 (D. Knuth, pers. comm., Jul. 11, 2025).
The Grabarchuk graph is implemented in the Wolfram Language as GraphData["GrabarchukGraph"].
D. Knuth (pers. comm., Jul. 11, 2025) posed the question whether the Grabarchuk graph has a Hamilton decomposition. Knuth (2025, p. 105) reports that E. Weisstein answered this in the affirmative on Aug. 21, 2025.
Pegg, E. Jr. Mathematical Games. Episode 32: "Unit Distance Graphs."
Aug. 21, 2025.