From the Coxeter Graph to the Klein Graph

نویسنده

  • Italo J. Dejter
چکیده

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.

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عنوان ژورنال:
  • Journal of Graph Theory

دوره 70  شماره 

صفحات  -

تاریخ انتشار 2012