3-symmetric and 3-decomposable geometric drawings of Kn

Abstract

Even the most supercial 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 3-symmetric). And second, they all are 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 n 4 + Θ(n 3) crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the general upper bound for the rectilinear crossing number of K n to 0.380488 n 4 + Θ(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.

There is a unique crossing-minimal rectilinear drawing of K_18

“…The value of cr(K 18 ) = 1029 was established in [8]. Crossing-minimal rectilinear drawings of K n for this range of values of n can be found in [2] and [6]. In particular, from [6], we know that there are at least 37269 non-isomorphic crossing-minimal drawings of K 17 .…”

“…This is actually done in Section 4.1, where we give the algorithm to label the points in P . Indeed, the unambiguity in the labelling of the points in P is established in Proposition 4.3 (1), and in order to prove this we need to prove simultaneously Proposition 4.3 (2), which in particular implies Lemma 3.1 for the case in which pq is a (≤5)-edge.…”

“…Let θ denote the counterclockwise rotation of 2π/3 around the origin, and let W := {(−51, 113), (6,834), (16, 989), (18, 644), (18, 1068), (22, 211)}. From [2], we know that the 18-point set W ∪ θ(W ) ∪ θ 2 (W ) induces a crossing-minimal drawing of K 18 . See Figure 1 for an illustration of such a point set.…”

“…As in [2], if {A, B, C} is a 3-decomposition of Q, we define two types of edges. Let p and q be distinct points of Q.…”

An Upper Bound Asymptotically Tight for the Connectivity of the Disjointness Graph of Segments in the Plane

“…where ρ is a 2π/3 clockwise rotation around a suitable point in the plane. The relationship between the concept of 3-symmetry and several variants of crossing number have been investigated by a number of authors [19][20][21][22]. If X is finite and |X| ≡ 0 (mod 3), then we say that X is almost 3-symmetric if X contains a subset X with at most two elements such that X \ X is 3-symmetric.…”