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dc.contributor.authorKroeker, Matthew Eliot 19:27:39 (GMT) 19:27:39 (GMT)
dc.description.abstractIn 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.publisherUniversity of Waterlooen
dc.subjectgraph theoryen
dc.subjectgraph colouringen
dc.titleSparsity in Critical Graphs with Small Clique Numberen
dc.typeMaster Thesisen
dc.pendingfalse and Optimizationen and Optimizationen of Waterlooen
uws-etd.degreeMaster of Mathematicsen
uws.contributor.advisorPostle, Luke
uws.contributor.affiliation1Faculty of Mathematicsen

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