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dc.contributor.authorSmith-Roberge, Evelyne
dc.date.accessioned2018-08-22 19:07:25 (GMT)
dc.date.available2018-08-22 19:07:25 (GMT)
dc.date.issued2018-08-22
dc.date.submitted2018-08
dc.identifier.urihttp://hdl.handle.net/10012/13643
dc.description.abstractLet $H$ be a graph. A graph $G$ is $H$-critical if every proper subgraph of $G$ admits a homomorphism to $H$, but $G$ itself does not. In 1981, Jaeger made the following conjecture concerning odd-cycle critical graphs: every planar graph of girth at least $4t$ admits a homomorphism to $C_{2t+1}$ (or equivalently, has a $\tfrac{2t+1}{t}$-circular colouring). The best known result for the $t=3$ case states that every planar graph of girth at least 18 has a homomorphism to $C_7$. We improve upon this result, showing that every planar graph of girth at least 16 admits a homomorphism to $C_7$. This is obtained from a more general result regarding the density of $C_7$-critical graphs. Our main result is that if $G$ is a $C_7$-critical graph with $G \not \in \{C_3, C_5\}$, then $e(G) \geq \tfrac{17v(G)-2}{15}$. Additionally, we prove several structural lemmas concerning graphs that are $H$-critical, when $H$ is a vertex-transitive non-bipartite graph.en
dc.language.isoenen
dc.publisherUniversity of Waterlooen
dc.subjecthomomorphismen
dc.subjectcircular colouringen
dc.subjectgraph theoryen
dc.subjectpotential methoden
dc.subjectdischargingen
dc.subjectcircular flow conjectureen
dc.subjectodd cycleen
dc.subjectcriticalen
dc.titleDensity and Structure of Homomorphism-Critical Graphsen
dc.typeMaster Thesisen
dc.pendingfalse
uws-etd.degree.departmentCombinatorics and Optimizationen
uws-etd.degree.disciplineCombinatorics and Optimizationen
uws-etd.degree.grantorUniversity of Waterlooen
uws-etd.degreeMaster of Mathematicsen
uws.contributor.advisorPostle, Luke
uws.contributor.affiliation1Faculty of Mathematicsen
uws.published.cityWaterlooen
uws.published.countryCanadaen
uws.published.provinceOntarioen
uws.typeOfResourceTexten
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen


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