UWSpace is currently experiencing technical difficulties resulting from its recent migration to a new version of its software. These technical issues are not affecting the submission and browse features of the site. UWaterloo community members may continue submitting items to UWSpace. We apologize for the inconvenience, and are actively working to resolve these technical issues.
 

Uniqueness and Complexity in Generalised Colouring

Thumbnail Image

Files

afarrugia2003.pdf (721.23 KB)

Date

2003

Authors

Farrugia, Alastair

Journal Title

Journal ISSN

Volume Title

Publisher

University of Waterloo

Abstract

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colouring perspective. Our main results are: · Additive induced-hereditary families are uniquely factorisable into irreducible families. · If <i>P</i> and <i>Q</i> are additive induced-hereditary graph families, then (<i>P</i>,<i>Q</i>)-COLOURING is NP-hard, with the exception of GRAPH 2-COLOURING. Moreover, with the same exception, (<i>P</i>,<i>Q</i>)-COLOURING is NP-complete iff <i>P</i>- and <i>Q</i>-RECOGNITION are both in NP. This proves a 1997 conjecture of Kratochvíl and Schiermeyer. We also provide generalisations to somewhat larger families. Other results that we prove include: · a characterisation of the minimal forbidden subgraphs of a hereditary property in terms of its minimal forbidden induced-subgraphs, and <i>vice versa</i>; · extensions of Mihók's construction of uniquely colourable graphs, and Scheinerman's characterisations of compositivity, to disjoint compositive properties; · an induced-hereditary property has at least two factorisations into arbitrary irreducible properties, with an explicitly described set of exceptions; · if <i>G</i> is a generating set for <i>A</i> &#959; <i>B</i>, where <i>A</i> and <i>B</i> are indiscompositive, then we can extract generating sets for <i>A</i> and <i>B</i> using a <i>greedy algorithm</i>.

Description

Keywords

Mathematics, graph, graph property, vertex partition, graph colouring, hereditary, induced-hereditary, compositive, complexity, uniquely colourable graphs, uniquely partitionable graphs, unique factorisation

LC Keywords

Citation

URI

http://hdl.handle.net/10012/1018

Collections

Theses
Combinatorics and Optimization