Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 3--symmetric and 3--decomposable drawings of $K_n$ (extended version)   Even 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
$0.380029\binom{n}{4}+\Theta(n^3)$ 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
$0.380488\binom{n}{4}+\Theta(n^3)$. We also give explicit 3-symmetric and
3-decomposable constructions for $n<100$ that are at least as good as those
previously known.

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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: ao-sci