The Haugland graphs are the three unit-distance graphs 
,
, and 
used by Haugland (2026) in the construction of a Moser
spindle-free graph with chromatic number
5. They are called Haugland graphs in this work.
| graph | vertex count | edge count | role |
 | 740 | 3985 | initial graph |
 | 1066 | 6264 | intermediate graph |
 | 2131 | 12530 | final 5-chromatic graph |
The vertices of 
are exact partial sums along
231 polygonal paths whose steps are selected from 84 directed
unit vectors obtained from a 21-vertex unit-distance
graph with sevenfold rotational symmetry.
Exact rotations and translations
of two copies of 
produce 
.
An exact rotation and translation of a second copy of 
, together with the original, produces 
.
The three graphs will be implemented in a future version of the Wolfram Language as GraphData["HauglandGraph740"], GraphData["HauglandGraph1066"], and GraphData["HauglandGraph2131"], respectively.
All three graphs are unit-distance graphs realized in the Euclidean plane, but are nonplanar
graphs. The final graph 
has chromatic number
5 and contains no Moser spindle (Haugland 2026).
It is the smallest known unit-distance graph
with chromatic number 5 subject to the latter
restriction, while the smallest known without that restriction remains the 509-vertex
Parts graph.
