@MISC{Dejter_fromthe, author = {Italo J. Dejter}, title = {From the Coxeter graph to the Klein graph}, year = {} }

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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 ⃗ C7 and the 2-arcs ⃗ P3 that tightly fasten those ⃗ C7 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 C7 and the 1-paths P2 that tightly fasten those C7 in Γ ′. This yields an embedding of Γ ′ into a 3-torus T3 which forms the Klein map of Coxeter notation (7, 3)8. The dual graph of Γ ′ in T3 is the distance-regular Klein quartic graph, with corresponding dual map of Coxeter notation (3,7)8.