2010/02/09 by Italo J. Dejter, Dejter, Italo J.
Mathematics · #05B30 #05C20 #05C38 #05C62 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics #Combinatorics (math.CO) #Coxeter element #Coxeter group #Discrete mathematics #Dual graph #FOS: Mathematics #Finite Group Theory Research #Graph #Mathematics #Notation #Planar graph #Pure mathematics #Quartic function #Vertex (graph theory) #math.CO #msc:05B30 #msc:05C20 #msc:05C38 #msc:05C62
paper · pdf · doi:10.48550/arxiv.1002.1960
published in arXiv (Cornell University) (Cornell University) · 9 pages, 3 figures, 3 tables
openalex publication_date 2010/02/09 · arxiv created 2012/06/10 · arxiv updated 2012/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the 56-vertex Klein cubic graph \G' can be obtained from the 28-vertex Coxeter cubic graph \G by 'zipping' adequately the squares of the 24 7-cycles of \G endowed with an orientation obtained by considering \G as a \mathcal C-ultrahomogeneous digraph, where \mathcal C is the collection formed by both the oriented 7-cycles C7 and the 2-arcs P3 that tightly fasten those C7 in \G. In the process, it is seen that \G' is a \mathcal C'-ultrahomogeneous (undirected) graph, where \mathcal C' is the collection formed by both the 7-cycles C7 and the 1-paths P2 that tightly fasten those C7 in \G'. This yields an embedding of \G' into a 3-torus T3 which forms the Klein map of Coxeter notation (7,3)8. The dual graph of \G' in T3 is the distance-regular Klein quartic graph, with corresponding dual map of Coxeter notation (3,7)8.