A Systematic Approach to Crossing Numbers of Cartesian Products with Paths

TL;DR

The paper addresses determining the crossing numbers $cr(G\Box P_n)$ for Cartesian products of small graphs $G$ with long paths $P_n$. It introduces binary-weighted capacity-constrained CNP (BCCNP) and reduces lower-bound proofs to solving small subproblems $cr_a(G,d,\cdot)$ on $G\Box P_d$ with $d$ small, employing $a$-restricted drawings and crossing bands along with plus-forces and star-forces. The authors implement an ILP-based BCCNP solver to certify base cases and force computations, achieving 128 successful determinations across 133 non-isomorphic connected graphs of orders five or six (including 60 previously unresolved cases) and providing matching upper-bound constructions. They also validate results with certificates and outline extensions to other graph products and cycles, showing a scalable framework for systematic crossing-number analysis. This work substantially broadens the known landscape of cr$(G\Box P_n)$ cases and offers practical, verifiable methodology for future investigations in graph drawing theory.

Abstract

Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former is usually demonstrated by providing a suitable drawing procedure, while the latter often requires substantial theoretical arguments. Many such papers have been published, which typically focus on just one or two small graphs at a time, and use ad hoc arguments specific to those graphs. We propose a general approach which, when successful, establishes the required lower bound. This approach can be applied to the Cartesian product of any graph with arbitrarily large paths, and in each case involves solving a modified version of the crossing number problem on a finite number (typically only two or three) of small graphs. We demonstrate the potency of this approach by applying it to Cartesian products involving all 133 graphs $G$ of orders five or six, and show that it is successful in 128 cases. This includes 60 cases which a recent survey listed as either undetermined, or determined only in journals without adequate peer review.

Figures (6)

• Figure 1: Two drawings of $K_4 \Box P_4$. The $K_4$-copies are solid red; the path edges are dotted blue.
• Figure 2: Crossing bands $A^d_i$ for $G\Box P_d$, visually highlighted as colored diagonal stripes. A colored cell represents crossings between edges of the subgraphs corresponding to the row and the column, respectively. The number "$i$" in a cell indicates that these crossings are contained in $A^d_i$. Lighter shaded cells indicate redundant associations. In contrast, the restriction of an $a$-restricted drawing on copy $G^i$ sums over all crossings in row and column $G^i$; this is marked in bold black lines for $G^3$. Note that the latter contains a very different set of crossings than does $A^d_3$ (blue cells labeled "3").
• Figure 3: Crossing bands considered in a (plus-)force triple: (left) front, (right) middle and end.
• Figure A1: Results for 5-vertex graphs. We omit the planar instances $i = 1, 8$.
• Figure A2: Results for 6-vertex graphs

Theorems & Definitions (7)

• Definition 1
• Lemma 2
• Lemma 3
• Theorem 4
• Lemma 5
• Theorem 6
• Theorem 7