On Pappus, Desargues and other C-UH graphs

The notion of a C-ultrahomogeneous (or C-UH) graph, due to D. Isaksen et al., is adapted for digraphs and applied to the cubic distancetransitive graphs, considered as digraphs by replacing each edge by a pair of oppositely oriented arcs, with C formed by shortest oriented cycles and (k−1)-arcs with k = arc-transitivity. Moreover, (k−1)-powers of shortest cycles taken with orientation assignments that make these graphs become C-UH digraphs are ‘zipped’, or ‘fastened’, into C-UH graphs. In this note, this procedure is completed for the Desargues graph and adapted for the Pappus graph. Also, generalizing on the ‘zipping’ result over the Desargues graph, it can be concluded that the line graph L(Kn), with n ≥ 4, is a tightly fastened {Kn−1, K3}K2 -UH graph with n copies of Kn−1 and (