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From the Coxeter Graph to the Klein Graph
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AbstractWe 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 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. © 2011 Wiley Periodicals, Inc. J Graph Theory
Title: From the Coxeter Graph to the Klein Graph
Description:
AbstractWe 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 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.
© 2011 Wiley Periodicals, Inc.
J Graph Theory.
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