On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs

TL;DR

This paper studies the rectilinear crossing numbers $\overline{\operatorname{cr}}(G)$ of complete balanced multipartite graphs $K_n^r$ and layered graphs $L_n^r$. It develops two bounding strategies: (i) random embeddings of $K_n^r$ or $L_n^r$ into an optimal drawing of $K_n$ and (ii) planted drawings formed by replacing each vertex of a seed drawing with a cluster of collinear vertices. The authors derive explicit upper bounds, e.g., $cr(K_n^r) \le E(cr(D))$ leading to $cr(K_n^r) \le \frac{1}{16} \left(\frac{r-1}{r}\right)^2 \left(\frac{n^4}{4}-\frac{3n^3}{2}\right)+O(n^2)$ and $\overline{cr}(K_n^r) \le \frac{\overline{q}}{4!} \left(\frac{r-1}{r}\right)^2 n^4+o(n^4)$, with $cr(L_n^r) \le \frac{(r-1)^2}{16 r^4} n^4+O(n^3)$; planted methods yield improved leading-term bounds for certain $r$, notably for layered graphs. The appendix provides lower-bound results and concrete seed drawings, including a rectilinear $K_{24}^4$ with 2033 crossings, illustrating practically competitive seed-based constructions. Overall, the work advances upper-bound techniques and seed-based methods for multipartite and layered graphs, contributing to the broader understanding of rectilinear crossing behavior and its asymptotics.

Abstract

A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let $n \ge r$ be positive integers. The graph $K_n^r$, is the complete $r$-partite graph on $n$ vertices, in which every set of the partition has at least $\lfloor n/r \rfloor$ vertices. The layered graph, $L_n^r$, is an $r$-partite graph on $n$ vertices, in which for every $1\le i \le r-1$, all the vertices in the $i$-th partition are adjacent to all the vertices in the $(i+1)$-th partition. In this paper, we give upper bounds on the rectilinear crossing numbers of $K_n^r$ and~$L_n^r$.

Figures (5)

• Figure 1: An example of Hill's drawings of $K_{10}$, where here for convenience only the edges of one vertex are drawn. Left: the drawing on a cylinder. Right: an equivalent representation of Hill's drawings via concentric circles.
• Figure 2: A drawing of $K_8^4$ with 6 crossings (left) and $K_9^4$ with 15 crossings (right).
• Figure 3: The seeds for the planted drawings of $K_n^2$ and $K_n^3$
• Figure 4: The rectilinear $D_{6}$ drawing of $L_{12}^6$
• Figure 5: The vertices of a rectilinear drawing of $K_{24}^4$

Theorems & Definitions (16)

• Conjecture 1: Harary-Hill guy_conj
• Conjecture 2: Zarankiewicz
• Conjecture 3
• Proposition 4
• Theorem 5
• Lemma 5
• Theorem 6
• Theorem 7
• Corollary 7
• Conjecture 8