On the geometric $k$-colored crossing number of $K_n$

TL;DR

The paper addresses the geometric $k$-colored crossing number $\overline{\overline{\text{cr}}}_k(K_n)$ and develops a generalized doubling construction to generate large-$n$ straight-line $k$-edge-colored drawings from small instances with few monochromatic crossings. It derives a recurrence yielding a closed-form asymptotic expression $cr_k(P_t;\chi_t)=\alpha 2^{4t}+\beta 2^{3t}+\gamma 2^{2t}+\delta 2^{t}$, where the dominant coefficient $\alpha$ is minimized via a per-vertex, minimum-weight $P_0$-saturating matching on a derived bipartite graph, computable in polynomial time. For $k=2$ it improves the known bound to $\overline{\overline{\text{cr}}}_2<0.11750015$, and for $k=3$–$10$ it uses MAX-$k$-CUT heuristics on crossing graphs to obtain new colorings $(P_k,\chi_k)$ and refined point sets, achieving upper bounds that beat the $k$-page book bound by roughly a factor of 3. The work advances understanding of geometric $k$-colored crossing numbers and suggests future directions, including stronger lower bounds via $\ell$-edges and non-convex configurations to extend the approach to larger $k$.

Abstract

We study the \emph{geometric $k$-colored crossing number} of complete graphs $\overline{\overline{\text{cr}}}_k(K_n)$, which is the smallest number of monochromatic crossings in any $k$-edge colored straight-line drawing of $K_n$. We substantially improve asymptotic upper bounds on $\overline{\overline{\text{cr}}}_k(K_n)$ for $k=2,\ldots, 10$ by developing a procedure for general $k$ that derives $k$-edge colored drawings of $K_n$ for arbitrarily large $n$ from initial drawings with a low number of monochromatic crossings. We obtain the latter by heuristic search, employing a \textsc{MAX-$k$-CUT}-formulation of a subproblem in the process.

Figures (2)

• Figure 1: One step in the doubling procedure at a vertex $p$. Matching edges are drawn bold and with an arrowhead. Dashed lines are the extensions of matching edges along which the vertices are split.
• Figure 2: The four cases we use to count the number of crossings after one step of the doubling construction.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 1
• proof
• Claim 1
• proof : Proof of \ref{['claim:one_step_crossings']}
• Lemma 1
• proof