The 3-symmetric Pseudolinear Crossing Number of $K_{33}$

TL;DR

This work determines the exact value of the 3-symmetric pseudolinear and rectilinear crossing numbers for $K_{33}$ by encoding point configurations with allowable sequences and exploiting 3-decomposability and 3-symmetry. The authors derive tight bounds on the relevant transposition counts and show that any potential 3-symmetric halfperiod cannot realize a crossing number below the known bound, culminating in the equality $\operatorname{sym}-\widetilde{\operatorname{cr}}_3(K_{33}) = \operatorname{sym}-\overline{\operatorname{cr}}_3(K_{33}) = 14{,}634$. The analysis hinges on a careful partition of transpositions into monochromatic and bichromatic types and a contradiction argument based on the $16$th level halvings. The result advances understanding of the structure of optimal 3-symmetric drawings and links rectilinear and pseudolinear optimality in the $K_{33}$ case.

Abstract

We show that the 3-symmetric rectilinear and the 3-symmetric pseudolinear crossing numbers of $K_{33}$ are equal. Specifically, we prove that $\operatorname{sym}-\overline{\operatorname{cr}}_3(K_{33}) = 14 634 = \operatorname{sym}-\widetilde{\operatorname{cr}}_3(K_{33})$.

Figures (3)

• Figure 1: A 9-points set which is $3$-decomposable.
• Figure 2: The underlying set of points of a $3$-symmetric drawing of $K_{12}$ (left) and a $6$-symmetric drawing of $K_{24}$ (right).
• Figure 3: The underlying set of points $P$ of an optimal 3-symmetric 3-decomposable rectilinear drawing of $K_{33}$ with seed set $Q$ and center of rotation at $(0,0)$. Namely, $\operatorname{sym}-\overline{\operatorname{cr}}_3(P)=14~634$.

Theorems & Definitions (42)

• Theorem 1
• Remark 4
• Proposition 5
• proof
• Theorem 6: Main Theorem, Cetina et al. cetina2011point
• Remark 8
• Remark 9
• Proposition 10: Claim 1, Ábrego et al. abrego20103sym
• Corollary 11
• Corollary 12