3--symmetric and 3--decomposable drawings of $K_n$ (extended version)

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dc.creatorÁbrego, B.
dc.creatorCetina, M.
dc.creatorFernández--Merchant, S.
dc.creatorLeaños, J.
dc.creatorSalazar, G.
dc.date2008-04-30
dc.date2008-05-08
dc.date.accessioned2026-07-07T09:37:30Z
dc.date.available2026-07-07T09:37:30Z
dc.descriptionEven the most superficial glance at the vast majority of crossing-minimal geometric drawings of $K_n$ reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply {\em 3-symmetric}) . And second, they all are {\em 3-decomposable}, that is, there is a triangle $T$ enclosing the drawing, and a balanced partition $A, B, C$ of the underlying set of points $P$, such that the orthogonal projections of $P$ onto the sides of $T$ show $A$ between $B$ and $C$ on one side, $B$ between $A$ and $C$ on another side, and $C$ between $A$ and $B$ on the third side. In fact, we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric optimal constructions for all $n$ multiple of 3. In this paper, we show that any 3-decomposable geometric drawing of $K_n$ has at least $0.380029\binom{n}{4}+Θ(n^3)$ crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the {\em general} upper bound for the rectilinear crossing number of $K_n$ to $0.380488\binom{n}{4}+Θ(n^3)$. We also give explicit 3-symmetric and 3-decomposable constructions for $n<100$ that are at least as good as those previously known.
dc.descriptionAdded new content
dc.identifierhttps://arxiv.org/abs/0805.0016
dc.identifierhttp://arxiv.org/abs/0805.0016
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/160479
dc.subjectCombinatorics
dc.subject05C10
dc.title3--symmetric and 3--decomposable drawings of $K_n$ (extended version)
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