A Systematic Approach to Crossing Numbers of Cartesian Products with Paths

TL;DR

This paper introduces a general method for determining crossing numbers of Cartesian products of small graphs with large paths, successfully applying it to most cases of graphs with five or six vertices.

Contribution

It presents a systematic approach that establishes lower bounds for crossing numbers by reducing the problem to small graphs, covering all graphs of order five or six.

Findings

Successfully applied to 128 out of 133 graphs of order five or six.

Resolved many previously undetermined crossing number cases.

Provides a unified framework for crossing number analysis of Cartesian products.

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.