Pappus-Desargues digraph confrontation

zbMATH Open1300.05115arXiv0904.1096MaRDI QIDQ2875895

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Publication date: 12 August 2014

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 $mathcalP$ and $mathcalD$ (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 $mathcalP$ and opposite for $mathcalD$, now into 2 disjoint copies of their corresponding Menger graphs. Here, $mathcalP$ is the unique cubic distance-transitive (or CDT) graph with the concurrent-reattachment behavior while $mathcalD$ is one of 7 CDT graphs with the opposite-reattachment behavior, that include the Coxeter graph. Thus, $mathcalP$ and $mathcalD$ confront each other in these respects, obtained via $mathcalC$-ultrahomogeneous graph techniques [3,4] that allow to characterize the obtained reattachment Menger graphs in the same terms.

Full work available at URL: https://arxiv.org/abs/0904.1096

zbMATH Keywords

Coxeter graphLevi graphsKlein graph

Mathematics Subject Classification ID

Directed graphs (digraphs), tournaments (05C20) Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) (05C60)

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