dc.contributor.author Smith-Roberge, Evelyne dc.date.accessioned 2018-08-22 19:07:25 (GMT) dc.date.available 2018-08-22 19:07:25 (GMT) dc.date.issued 2018-08-22 dc.date.submitted 2018-08 dc.identifier.uri http://hdl.handle.net/10012/13643 dc.description.abstract Let $H$ 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.iso en en dc.publisher University of Waterloo en dc.subject homomorphism en dc.subject circular colouring en dc.subject graph theory en dc.subject potential method en dc.subject discharging en dc.subject circular flow conjecture en dc.subject odd cycle en dc.subject critical en dc.title Density and Structure of Homomorphism-Critical Graphs en dc.type Master Thesis en dc.pending false uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree.discipline Combinatorics and Optimization en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Master of Mathematics en uws.contributor.advisor Postle, Luke uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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