From the Coxeter Graph to the Klein Graph

✍ Scribed by Italo J. Dejter

• Book ID: 102339850
• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 208 KB
• Volume: 70
• Category: Article
• ISSN: 0364-9024
• DOI: 10.1002/jgt.20597

✦ Synopsis

Abstract

We show that the 56‐vertex Klein cubic graph Γ′ can be obtained from the 28‐vertex Coxeter cubic graph Γ by “zipping” adequately the squares of the 24 7‐cycles of Γ endowed with an orientation obtained by considering Γ as a 𝒞‐ultrahomogeneous digraph, where 𝒞 is the collection formed by both the oriented 7‐cycles and the 2‐arcs that tightly fasten those in Γ. In the process, it is seen that Γ′ is a 𝒞′‐ultrahomogeneous (undirected) graph, where 𝒞′ is the collection formed by both the 7‐cycles C~7~ and the 1‐paths P~2~ that tightly fasten those C~7~ in Γ′. This yields an embedding of Γ′ into a 3‐torus T~3~ which forms the Klein map of Coxeter notation (7, 3)~8~. The dual graph of Γ′ in T~3~ is the distance‐regular Klein quartic graph, with corresponding dual map of Coxeter notation (3, 7)~8~. © 2011 Wiley Periodicals, Inc. J Graph Theory