From the Coxeter Graph to the Klein Graph

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 ‐ultrahomogeneous digraph, where 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 ′‐ultrahomogeneous (undirected) graph, where ′ 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