dc.contributor.author Kroeker, Matthew Eliot dc.date.accessioned 2020-08-27 19:27:39 (GMT) dc.date.available 2020-08-27 19:27:39 (GMT) dc.date.issued 2020-08-27 dc.date.submitted 2020-08-17 dc.identifier.uri http://hdl.handle.net/10012/16178 dc.description.abstract In 1998, Reed conjectured that for every graph $G$, $\chi(G) \leq \lceil \frac{1}{2}(\Delta(G)+1+\omega(G)) \rceil$, and proved that there exists $\varepsilon > 0$ such that $\chi(G) \leq \lceil (1 - \varepsilon)(\Delta(G)+1) + \varepsilon \omega(G) \rceil$ for every graph $G$. Recently, much effort has been made to prove this result for increasingly large values of $\varepsilon$ in graphs with sufficiently large maximum degree. One of the main lemmas used in deriving these bounds states that graphs which are list-critical are sparse. This result generally follows by applying a sufficient condition for list colouring complete multipartite graphs with parts of bounded size, and until recently a theorem of Erd\H{o}s, Rubin and Taylor for list colouring complete multipartite graphs with parts of size at most two was used. The current bottleneck in bounding $\chi(G)$ for an improved value of $\varepsilon$ is the case of small clique number. We derive new density lemmas exploiting this case by showing that our graph is contained in a complete multipartite graph with many parts of size three. In order to list colour in this setting, we apply a theorem of Noel, West, Wu and Zhu, as well as our own unbalanced variant of this result. en dc.language.iso en en dc.publisher University of Waterloo en dc.subject graph theory en dc.subject graph colouring en dc.title Sparsity in Critical Graphs with Small Clique Number 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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