The Grötzsch graph is smallest triangle-free graph with chromatic number four. It is identical to the Mycielski graph of order four, and is implemented as GraphData["GrotztschGraph"]. It has 11 vertices and 20 edges. It is Hamiltonian, but nonplanar. It is illustrated above in a number of embeddings.
The plots above show the adjacency, incidence, and graph distance matrices for the Grötzsch graph.
The graph spectrum of the Grötzsch graph is 
.
de Grey (2026) considered a unit-distance embedding of the Grötzsch graph in three dimensions in his construction
of 5-chromatic, triangle-free, unit-distance
graph in 
,
though ended up using a different graph on 31 vertices. The embedding considered
by de Grey, together with a similar one based on a pentagram
instead of a pentagon, is illustrated above.
With 61 Vertices." Geombinatorics 35, 2026.Soifer,
A. 
th
Chromatic Numbers of the Grötzsch Graph." J. Graph Th. 21,
207-209, 1996.