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Symmetric Graphs from Polytopes of High Rank

par Mixer, Mark ;Schulte, Egon
Référence Graphs and combinatorics, 28, 6, page (843-857)
Publication Publié, 2012-11

Article révisé par les pairs

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Résumé :

Given a polytope P of rank 2n, the faces of middle ranks n - 1 and n constitute the vertices of a bipartite graph, the medial layer graph M(P) of P. The group D(P) of automorphisms and dualities of P has a natural action on this graph. We prove algebraic and combinatorial conditions on P that ensure this action is transitive on k-arcs in M(P) for some small k (in particular focussing on k = 3), and provide examples of families of polytopes that satisfy these conditions. We also examine how D(P) acts on the k-stars based at vertices of M(P) and describe self-dual regular polytopes (in particular those of rank 6) for which this action is transitive on the k-stars for small k. © 2011 Springer.