Pappus-Desargues digraph confrontation

TL;DR

This paper explores the reattachment properties of the Levi graphs of Pappus and Desargues configurations, revealing unique behaviors in their Menger graphs and employing ultrahomogeneous graph techniques.

Contribution

It introduces a novel analysis of the reattachment behavior of Pappus and Desargues Levi graphs, identifying their unique and shared properties using ultrahomogeneous graph methods.

Findings

Pappus graph exhibits concurrent-reattachment behavior.

Desargues graph shows opposite-reattachment behavior.

Both graphs are characterized as distance-transitive and ultrahomogeneous.

Abstract

Like the Coxeter graph became reattached into the Klein graph in [2], the Levi graphs of the $9_3$ and $10_3$ self-dual configurations, known as the Pappus and Desargues ($k$-transitive) graphs $\mathcal P$ and $\mathcal D$ (where $k=3$), also admit reattachments of the distance-$(k-1)$ graphs of half of their oriented shortest cycles via orientation assignments on their common $(k-1)$-arcs, concurrent for ${\mathcal P}$ and opposite for $\mathcal D$, now into 2 disjoint copies of their corresponding Menger graphs. Here, $\mathcal P$ is the unique cubic distance-transitive (or CDT) graph with the concurrent-reattachment behavior while $\mathcal D$ is one of 7 CDT graphs with the opposite-reattachment behavior, that include the Coxeter graph. Thus, $\mathcal P$ and $\mathcal D$ confront each other in these respects, obtained via $\mathcal C$-ultrahomogeneous graph techniques [3,4] that allow to characterize the obtained reattachment Menger graphs in the same terms.