Uniqueness and Complexity in Generalised Colouring

dc.contributor.authorFarrugia, Alastairen
dc.date.accessioned2006-08-22T14:25:13Z
dc.date.available2006-08-22T14:25:13Z
dc.date.issued2003en
dc.date.submitted2003en
dc.description.abstractThe 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>.en
dc.formatapplication/pdfen
dc.format.extent738537 bytes
dc.format.mimetypeapplication/pdf
dc.identifier.urihttp://hdl.handle.net/10012/1018
dc.language.isoenen
dc.pendingfalseen
dc.publisherUniversity of Waterlooen
dc.rightsCopyright: 2003, Farrugia, Alastair. All rights reserved.en
dc.subjectMathematicsen
dc.subjectgraphen
dc.subjectgraph propertyen
dc.subjectvertex partitionen
dc.subjectgraph colouringen
dc.subjecthereditaryen
dc.subjectinduced-hereditaryen
dc.subjectcompositiveen
dc.subjectcomplexityen
dc.subjectuniquely colourable graphsen
dc.subjectuniquely partitionable graphsen
dc.subjectunique factorisationen
dc.titleUniqueness and Complexity in Generalised Colouringen
dc.typeDoctoral Thesisen
uws-etd.degreeDoctor of Philosophyen
uws-etd.degree.departmentCombinatorics and Optimizationen
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

Files

Original bundle

Now showing 1 - 1 of 1
Loading...
Thumbnail Image
Name:
afarrugia2003.pdf
Size:
721.23 KB
Format:
Adobe Portable Document Format