Density and Structure of Homomorphism-Critical Graphs
dc.contributor.author | Smith-Roberge, Evelyne | |
dc.date.accessioned | 2018-08-22T19:07:25Z | |
dc.date.available | 2018-08-22T19:07:25Z | |
dc.date.issued | 2018-08-22 | |
dc.date.submitted | 2018-08 | |
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.identifier.uri | http://hdl.handle.net/10012/13643 | |
dc.language.iso | en | en |
dc.pending | false | |
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 |
uws-etd.degree | Master of Mathematics | en |
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.contributor.advisor | Postle, Luke | |
uws.contributor.affiliation1 | Faculty of Mathematics | en |
uws.peerReviewStatus | Unreviewed | en |
uws.published.city | Waterloo | en |
uws.published.country | Canada | en |
uws.published.province | Ontario | en |
uws.scholarLevel | Graduate | en |
uws.typeOfResource | Text | en |