We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 05 June 2012
The crossing number of a graph G is the smallest number of pairwise crossings of edges among all drawings of G in the plane. In the last decade, there has been significant progress on a true theory of crossing numbers. There are now many theorems on the crossing number of a general graph and the structure of crossing-critical graphs, whereas in the past, most results were about the crossing numbers of either individual graphs or the members of special families of graphs. This chapter highlights these recent advances and some of the open questions that they suggest.
Introduction
Historically, the study of crossing numbers has mainly been devoted to the computation of the crossing numbers of particular families of graphs. Given that we still do not know the crossing numbers for basic graphs such as the complete graphs and complete bipartite graphs, this is perhaps not surprising. However, a broader theory has recently begun to emerge. This theory has been used for computing crossing numbers of particular graphs, but has also promulgated open questions of its own. One of the aims of this chapter is to highlight some of these recent theoretical developments.
The study of crossing numbers began during the Second World War with Paul Turán. In [58], he tells the story of working in a brickyard and wondering about how to design an efficient rail system from the ‘kilns’ to the ‘storage yards’.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
