On the first two entries of the f-vectors of 6-polytopes
DOI:
https://doi.org/10.11575/cdm.v15i1.62385Abstract
In 1906, Steinitz gave a complete characterization of the first two entries of the $f$-vectors of $3$-polytopes, while Grünbaum obtained a similar result for $4$-polytopes in his well-known book published in 1967. Recently, Kusunoki and Murai and independently Pineda-Villavicencio, Ugon, and Yost completely determined the first two entries of the $f$-vectors of $5$-polytopes. This paper can be regarded as a continuation of their works for $6$-polytopes. To be more precise, let $k$ denote the number of vertices of a $6$-polytope. The aim of this paper is to show that, when the number of edges is greater than or equal to $\frac{7}{2}(k-1)$ and $k\ge 14$, we can completely characterize the first two entries of the $f$-vectors of $6$-polytopes. As a consequence, for $7\le k\le 15$ we also give a complete characterization of the first two entries of the $f$-vectors of $6$-polytopes except for three cases $(12, 39)$, $(13, 43)$, and $(15, 47)$.
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