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Browsing Combinatorics and Optimization by Supervisor "Postle, Luke"
Now showing items 17 of 7

Acyclic Colouring of Graphs on Surfaces
(University of Waterloo, 20180904)An acyclic kcolouring of a graph G is a proper kcolouring of G with no bichromatic cycles. In 1979, Borodin proved that planar graphs are acyclically 5colourable, an analog of the Four Colour Theorem. Kawarabayashi and ... 
Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm
(University of Waterloo, 20190809)Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most ... 
Cyclically 5Connected Graphs
(University of Waterloo, 20160829)Tutte's FourFlow Conjecture states that every bridgeless, Petersenfree graph admits a nowherezero 4flow. This hard conjecture has been open for over half a century with no significant progress in the first forty years. ... 
Density and Structure of HomomorphismCritical Graphs
(University of Waterloo, 20180822)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 oddcycle critical graphs: ... 
Fractional refinements of integral theorems
(University of Waterloo, 20210709)The focus of this thesis is to take theorems which deal with ``integral" objects in graph theory and consider fractional refinements of them to gain additional structure. A classic theorem of Hakimi says that for an ... 
Local Perspectives on Planar Colouring
(University of Waterloo, 20220809)In 1994, Thomassen famously proved that every planar graph is 5choosable, resolving a conjecture initially posed by Vizing and, independently, Erdos, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every ... 
Sparsity in Critical Graphs with Small Clique Number
(University of Waterloo, 20200827)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) ...