3-Symmetric Pseudolinear Crossing Number

• The paper shows that a 3-symmetric pseudolinear drawing of K33 has exactly 14,634 crossings, matching the rectilinear case.
• The methodology uses allowable sequences and critical transposition counts under symmetry constraints to derive tight crossing number bounds.
• The study highlights how 3-symmetry imposes rigid structure on graph drawings, limiting pseudolinear flexibility and informing combinatorial geometry.

The 3-symmetric pseudolinear crossing number is a graph drawing invariant that measures the minimum number of edge crossings in a complete graph $K_n$ when its drawing in the plane is both 3-symmetric (invariant under $120^\circ$ rotation) and pseudolinear (edges extended to form a pseudoline arrangement). This quantity emerges at the intersection of graph drawing theory and combinatorial geometry, particularly for the rectilinear and pseudolinear crossing numbers constrained by rotational symmetry. The crossing number hierarchy and the interaction with symmetry constraints have implications for extremal combinatorics, geometric graph theory, and the analysis of allowable sequences.

1. Fundamental Definitions and Crossing Number Hierarchy

Consider a graph $G$ and the following crossing number variants:

• The crossing number $\mathrm{cr}(G)$ is the minimum number of pairwise edge crossings in any drawing of $G$ in the plane.
• The rectilinear crossing number $\overline{\mathrm{cr}}(G)$ restricts to drawings where vertices are in general position and edges are straight-line segments.
• The pseudolinear crossing number $\widetilde{\mathrm{cr}}(G)$ considers drawings where each edge extends to a pseudoline forming an arrangement in the projective plane.

Generally,

$$\mathrm{cr}(G) \leq \widetilde{\mathrm{cr}}(G) \leq \overline{\mathrm{cr}}(G).$$

Define a $k$-symmetric drawing of $G$ as one invariant under a rotation of order $k$. For integer $k \geq 2$,

• $\mathit{sym}\,\overline{\mathrm{cr}_k}(G)$ is the minimum number of crossings in a $k$-symmetric rectilinear drawing.
• $\mathit{sym}\,\widetilde{\mathrm{cr}_k}(G)$ is the minimum number of crossings in a $k$-symmetric pseudolinear drawing.

The symmetry-constrained crossing numbers satisfy: $$\mathit{sym}\,\mathrm{cr}_k(G)\leq \mathit{sym}\,\widetilde{\mathrm{cr}_k}(G)\leq \mathit{sym}\,\overline{\mathrm{cr}_k}(G).$$

2. Central Results for $K_{33}$

For the complete graph $K_{33}$,

$$\mathit{sym}\,\overline{\mathrm{cr}_3}(K_{33}) = 14\,634 = \mathit{sym}\,\widetilde{\mathrm{cr}_3}(K_{33})$$

This asserts that the best achievable crossing number in a 3-symmetric pseudolinear arrangement of $K_{33}$ equals that in the rectilinear case, demonstrating no advantage from the added flexibility of pseudolines (Martínez et al., 14 Jan 2026).

3. Allowable Sequences and Crossing Count Formula

Pseudolinear drawings of $K_n$ correspond to allowable sequences—ordered lists of permutations where consecutive permutations differ by adjacent transpositions, and each pair transposes exactly once. For $n$ odd, the number of crossings in such a halfperiod $\Pi$ can be computed via critical transpositions: Let $N_{\leq k}(\Pi)$ denote the number of adjacent transpositions in positions $i$ with $i\leq k$ or $i\geq n-k$ across $\Pi$. For $n=33$,

$$\mathrm{crossings}(\Pi) = \sum_{k=1}^{16} (32 - 2k) N_{\leq k}(\Pi) - 3960$$

Lower bounds (Lovász–Abrego–Fernández‐Merchant) on $N_{\leq k}(\Pi)$ constrain possible drawings. In the 3-symmetric context, each $N_{\leq k}(\Pi)$ is a multiple of 3. The minimal vector for $K_{33}$ is: $$(N_{\leq 1},\dots,N_{\leq 16}) = (3,9,18,30,45,63,84,108,135,165,198,237,282,333,399,528)$$ This yields a hypothetical crossing count of $14\,628$—but this value is shown to be unattainable.

4. Tightness, Decomposability, and Structure Theorems

Any 3-symmetric halfperiod for $K_{33}$ achieving fewer than $14\,634$ crossings must realize the minimal vector above (Proposition 3.4). Theorem 3.5 of Cetina et al. states that an $n$-point set with $N_{\leq k}(\Pi)=3\binom{k+2}{2}$ for all $k<11$ is 3-decomposable: its points partition into parts $A,B,C$ satisfying a cyclic projection property. This structural constraint enables finer analysis of critical transpositions and demonstrates that any optimal sequence is 3-decomposable.

5. Monochromatic and Bichromatic Critical Transpositions

Within a 3-decomposable halfperiod, critical transpositions split into:

• Monochromatic: between points from the same part (e.g., $A,A$),
• Bichromatic: between points from different parts.

Proposition 4.1 provides explicit formulas for the number of bichromatic $(\leq k)$–transpositions, allowing deduction of monochromatic transpositions. By 3-symmetry (Corollary 4.5), counts split evenly among $A,A$, $B,B$, $C,C$. For $k=16$, $N_{16}^{aa}(\Pi)=32$ is required in the “ideal” case, but this is proven unattainable in Section 5 through combinatorial gating constraints and counting arguments.

6. Non-Existence of the Ideal Halfperiod

A detailed combinatorial argument (Propositions 5.2–5.13) labels elements of $A$ by their interactions with the middle third of $\Pi$ and establishes gating constraints. The analysis shows that no allowable sequence for $K_{33}$ can reach the hypothetical minimal counts for all $(\leq k)$–transpositions, establishing that $14\,628$ crossings is impossible for any 3-symmetric pseudolinear drawing.

7. Equivalence of 3-Symmetric Rectilinear and Pseudolinear Crossing Numbers for $K_{33}$

The existence of a concrete 3-symmetric rectilinear drawing of $K_{33}$ with $14\,634$ crossings, combined with the non-existence of a pseudolinear drawing reaching $14\,628$, yields

$$\mathit{sym}\,\overline{\mathrm{cr}_3}(K_{33}) = 14\,634 = \mathit{sym}\,\widetilde{\mathrm{cr}_3}(K_{33})$$

This demonstrates that for $K_{33}$, the flexibility of pseudolinear arrangements provides no advantage over straight-line drawings under 3-symmetry (Martínez et al., 14 Jan 2026). The minimal crossing number in both settings coincides, indicating a deep correspondence between geometric and combinatorial symmetry constraints in complete graph drawings.

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Table: Crossing Number Variants for $K_{33}$ (3-Symmetric Drawings)

Variant	Definition	Value for $K_{33}$
$\mathit{sym}\,\overline{\mathrm{cr}_3}$	Rectilinear, 3-symmetric	$14\,634$
$\mathit{sym}\,\widetilde{\mathrm{cr}_3}$	Pseudolinear, 3-symmetric	$14\,634$

The equality shown above underscores the rigidity imposed by 3-symmetry in both the geometric (rectilinear) and topological (pseudolinear) frameworks. A plausible implication is that, for certain $k$ and $n$, further investigations into the limitations of pseudolinear flexibility under symmetry may elucidate additional instances of crossing number coincidences.