Complete subgraphs of the graphs of convex polytopes |
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Authors: | S Gallivan ER Lockeberg P McMullen |
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Institution: | Department of Mathematics, University College London, Cower Street, London WC1E 6BT, England |
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Abstract: | It is shown that if three vertices of the graph G(l') of a convex 3-polytope P are chosen, then G(P) contains a refinement of the complete graph C4 on four vertices, for which the three chosen vertices are principal (that is, correspond to vertices of C4 in the refinement). In general, all four vertices may not be preassigned as principal. For dimensions d?4, simple (simplicial) d-polytopes are constructed whose graphs contain sets of three (four) vertices, which cannot all be principal in any refinement of C4+1. |
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