The triplex graph is the graph illustrated above in a number of embeddings. It is a cubic Hamiltonian graph on 12 vertices and 18 edges with graph crossing number 2, graph diameter 3, and girth 5. A construction of the triplex graph is given by Robertson et al. (2019). It is also the projective planar forbidden topological minor F13, appearing in projective-plane obstruction lists (Glover et al. 1979, Mohar and Thomassen 2001). Robertson et al. (2019) do not state the origin of the name, which may refer to the threefold structure of their graph embedding drawn in Fig. 3.
It is one of the two 12-vertex cubic graphs attaining the minimum graph transmission among connected cubic graphs on 12 vertices, the other being the twinplex graph. The triplex graph is implemented in the Wolfram Language as GraphData["TriplexGraph"].
