Now showing items 1-7 of 7

• #### Acyclic Colouring of Graphs on Surfaces ﻿

(University of Waterloo, 2018-09-04)
An acyclic k-colouring of a graph G is a proper k-colouring of G with no bichromatic cycles. In 1979, Borodin proved that planar graphs are acyclically 5-colourable, an analog of the Four Colour Theorem. Kawarabayashi and ...
• #### Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm ﻿

(University of Waterloo, 2019-08-09)
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 5-Connected Graphs ﻿

(University of Waterloo, 2016-08-29)
Tutte's Four-Flow Conjecture states that every bridgeless, Petersen-free graph admits a nowhere-zero 4-flow. This hard conjecture has been open for over half a century with no significant progress in the first forty years. ...
• #### Density and Structure of Homomorphism-Critical Graphs ﻿

(University of Waterloo, 2018-08-22)
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: ...
• #### Fractional refinements of integral theorems ﻿

(University of Waterloo, 2021-07-09)
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, 2022-08-09)
In 1994, Thomassen famously proved that every planar graph is 5-choosable, 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, 2020-08-27)
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) ...

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