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Descripción
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| In this paper we present three different results dealing with the number of $(\leq k)$-facetsof a set of points: \begin{enumerate} \item We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq k)$-edges for a fixed $0\leq k\leq \lfloor n/3 \rfloor -1$; \item We give a simple construction showing that the lower bound $3\binom{k+2}{2}+3\binom{k-\lfloor \frac{n}{3} \rfloor+2}{2}$ for the number of $(\leq k)$-edges of a planar point set appeared in~[Aichholzer et al. New lower bounds for the number of ($\leq k$)-edges and the rectilinear crossing number of $K_n$. {\em Disc. Comput. Geom.} 38:1 (2007), 1--14] is optimal in the range $\lfloor n/3 \rfloor \leq k \leq \lfloor 5n/12 \rfloor -1$; \item We show that for $k \leq \lfloor n/4 \rfloor -1$ the number of $(\leq k)$-facets of a set of $n$ points in general position in~$\mathbb{R}^3$ is at least $4\binom{k+3}{3}$, and that this bound is tight in that range. \end{enumerate} | |
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Internacional
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Si |
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JCR del ISI
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Si |
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Título de la revista
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EUR J COMBIN |
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ISSN
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0195-6698 |
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Factor de impacto JCR
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0,651 |
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Información de impacto
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Volumen
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-- |
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DOI
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Número de revista
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0 |
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