The tritetrahedral graph is the skeleton of the tritetrahedron, a concave polyhedron formed by joining three regular tetrahedra at their faces. It
is a 3-tree.

The Nechushtan graph, a 10-vertex 5-chromatic graph in 3 dimensions, consists of two copies of the tritetrahedral graph (de Grey
and Haugstrup 2022) and is denoted in de Grey (2020) and in de Grey and Haugstrup (2022).

The tritetrahedral graph is implemented in the Wolfram
Language as GraphData["TritetrahedralGraph"].

de Grey, A. "A Small 6-Chromatic Unit-distance Graph in ."
Geocombinatorics30, 5-13, 2020.de Grey, A. and Haugstrup,
A. "Two Small 6-Chromatic Unit-Distance Graphs in ." Geocombinatorics31, 110-115, 2022.