By Jacob Bean
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Applying a scan from left to right, one can show that then there must exist a (3, 4)-cut. ✷ We can relate r + c + d to the number of bends, similarly as in . Lemma 2. In a hexagonal drawing of a graph with n vertices and m edges, let b be the number of bends and let r, c, d be the number of truly used rows, columns and diagonals. Then b ≥ r + c + d − 3n + m. Combining this with r + c + d ≥ 18 give the lower bounds for K7 . Theorem 6. Any hexagonal drawing of K7 has at least 18 bends. Using K7 and other small graphs, we can build arbitrarily large graphs that also have a large lower bound on the number of bends, similarly as done in  for orthogonal 2D drawings and in  for orthogonal 3D drawings.
At the time the projection change occurs, r and s are adjacent in the ordering. and the ordering changes by transposing r and s. By keeping track of all permutations of the projections as L is rotated by 180o , we obtain a circular sequence ΠS . The crucial observation is that (≤ k)-sets are in one-to-one correspondence with (≤ k)-critical transpositions of ΠS . Observation 2 Let S be a set of n points in the plane in general position, and let k < n/2. Then η≤k (S) = χ≤k (ΠS ). Combining Theorem 1 and Observation 2 and recalling the deﬁnition of X≤k (n), one immediately obtains the following statement, obtained independently in  and .
25 (2001), 351–364. 22. A. W. Beineke, Topological graph theory. W. J. ), pp. 15–49. Academic Press (1978). edu Abstract. We prove that the number of distinct weaving patterns produced by n semi-algebraic curves in R 3 defined coordinate-wise by polynomials of degrees bounded by some constant d, is bounded by 2O(n log n) , where the implied constant in the exponent depends on d. This generalizes a similar bound obtained by Pach, Pollack and Welzl  for the case when d = 1. 1 Introduction In , Pach, Pollack and Welzl considered weaving patterns of n lines in R 3 and showed that asymptotically only a negligible fraction of possible weaving patterns are realizable by straight lines in R 3 (see Remark 2 below).