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- W1673739447 abstract "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 $0.380029binom{n}{4}+Theta(n^3)$ 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 $0.380488binom{n}{4}+Theta(n^3)$. We also give explicit 3-symmetric and 3-decomposable constructions for $n<100$ that are at least as good as those previously known." @default.
- W1673739447 created "2016-06-24" @default.
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- W1673739447 date "2008-04-30" @default.
- W1673739447 modified "2023-09-27" @default.
- W1673739447 title "3--symmetric and 3--decomposable drawings of $K_n$ (extended version)" @default.
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