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• W2029095819 abstract "In 1958, Hill conjectured that the minimum number of crossings in a drawing of $K_n$ is exactly $Z(n) = frac{1}{4} lfloorfrac{n}{2}rfloor leftlfloorfrac{n-1}{2}rightrfloor leftlfloorfrac{n-2}{2}rightrfloorleftlfloorfrac{n-3}{2}rightrfloor$. Generalizing the result by '{A}brego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $x$-monotone curves. In fact, our proof shows that the conjecture remains true for $x$-monotone drawings of $K_n$ in which adjacent edges may cross an even number of times, and instead of the crossing number we count the pairs of edges which cross an odd number of times. We further discuss a generalization of this result to shellable drawings, a notion introduced by '{A}brego et al. We also give a combinatorial characterization of several classes of $x$-monotone drawings of complete graphs using a small set of forbidden configurations. For a similar local characterization of shellable drawings, we generalize Carath'eodory's theorem to simple drawings of complete graphs." @default.
• W2029095819 created "2016-06-24" @default.
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• W2029095819 date "2014-11-15" @default.
• W2029095819 modified "2023-10-06" @default.
• W2029095819 title "Crossing Numbers and Combinatorial Characterization of Monotone Drawings of $$K_n$$ K n" @default.
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• W2029095819 doi "https://doi.org/10.1007/s00454-014-9644-z" @default.
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