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"].
."
Geocombinatorics 30, 5-13, 2020.de Grey, A. and Haugstrup,
A. "Two Small 6-Chromatic Unit-Distance Graphs in 
." Geocombinatorics 31, 110-115, 2022.