Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T16:53:54.235Z Has data issue: false hasContentIssue false

4 - Defining sets in combinatorics: a survey

Published online by Cambridge University Press:  05 May 2013

D.M. Donovan
Affiliation:
The University of Queensland
E.S. Mahmoodian
Affiliation:
The University of Queensland
C. Ramsay
Affiliation:
The University of Queensland
A.P. Street
Affiliation:
Sharif University of Technology
C. D. Wensley
Affiliation:
University of Wales, Bangor
Get access

Summary

Abstract

In a given class of combinatorial structures there may be many distinct objects with the same parameters. Two questions arise naturally.

  • Given two such objects, where and how do they differ?

  • How much of an individual object is needed to identify it uniquely?

These questions are obviously related, the first leading to the concept of a trade, and the second to that of a defining set. This survey deals with denning sets in block designs, graphs and some related structures. The corresponding trades in each structure are also discussed briefly.

Introduction

We start with a simple example. A graph G = (V, E) consists of a finite set V of elements called vertices, and a set E of unordered pairs of vertices, called edges. The complete graph on v vertices, Kv, is a graph in which all pairs of distinct vertices constitute edges, so that any graph on v or fewer vertices may be considered as a subgraph of Kv.

If v = 2n, then a one-factor of Kv is a set of n unordered pairs which between them contain each element of V precisely once. A defining set of a one-factor is a subset of its edges which uniquely identifies it. More generally, a perfect matching in a graph G on 2n vertices is a set of n edges incident with each vertex of V.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

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.

Available formats
×

Save book 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 Dropbox.

Available formats
×

Save book to Google Drive

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.

Available formats
×