Drawing Hamiltonian Cycles with No Large Angles
Dumitrescu, Adrian and Pach, János and Tóth, Géza (2010) Drawing Hamiltonian Cycles with No Large Angles. In: Graph Drawing 17th International Symposium, GD 2009, September 22-25, 2009, Chicago, IL, USA , pp. 3-14 (Official URL: http://dx.doi.org/10.1007/978-3-642-11805-0_3).
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Let n ≥ 4 be even. It is shown that every set S of n points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of n straight line edges such that the angle between any two consecutive edges is at most 2π/3. For n = 4 and 6, this statement is tight. It is also shown that every even-element point set S can be partitioned into at most two subsets, S1 and S2 , each admitting a spanning tour with no angle larger than π/2. Fekete and Woeginger conjectured that for suﬃciently large even n, every n-element set admits such a spanning tour. We conﬁrm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by ﬁnitely many rectiﬁable curves: for any ε > 0, these sets almost surely admit a spanning tour with no angle larger than ε.
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