Pappus and Desargues graph $\mathcal C$-homogeneity
Italo Jose Dejter · arXiv (Cornell University) · 2009
A notion of $\mathcal C$-ultrahomogeneous (or $\mathcal C$-UH) graph due to D. Isaksen et al. is adapted to digraphs and applied to the Pappus graph ${\mathcal P}$ and Desargues graph ${\mathcal D}$ considered as digraphs with $\mathcal C$ formed by oriented cycles $\vec{C}_6$ and 2-arcs $\vec{P}_3$. Distance-2 graphs of cycles $C_6$ in ${\mathcal D}$ (resp. $\mathcal P$) are taken with orientation assignments opposite (concordant) on common paths $P_3$, making it appear as a $\{\vec{C}_6\}_{\vec{P}_3}$-UH ($\{\vec{C}_6\}^{\vec{P}_3}$-UH) digraph. In $\mathcal D$, the $\vec{C}_6$\thinspace s are `zipped' (or `fastened') into a $\{K_4,K_3\}_{K_2}$-UH graph formed by two disjoint copies of $L(K_5)$. This `fastening' yields an infinite nested sequence of geometric realizations of $L(K_5)$ by means of tetrahedron barycenters, and adapted for ${\mathcal P}$ towards a slightly weaker dual $\mathcal C$-homogeneity characterization of $\mathcal P$ in terms of triangles. Finally, generalizing on the `fastening' result for ${\mathcal D}$, it is seen that $L(K_n)$, ($n\geq 4$), is a tightly fastened $\{K_{n-1},K_3\}_{K_2}$-UH graph having $n$ copies of $K_{n-1}$ and ${n\choose 3}$ copies of $K_3$.