Browsing Theses by Subject "graph colouring"
Now showing items 1-5 of 5
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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 ... -
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 ... -
On Finding Large Cliques when the Chromatic Number is close to the Maximum Degree
(University of Waterloo, 2021-12-23)We prove that every graph G with chromatic number χ(G) = ∆(G) − 1 and ∆(G) ≥ 66 contains a clique of size ∆(G) − 17. Our proof closely parallels a proof from Cranston and Rabern, who showed that graphs with χ = ∆ and ∆ ≥ ... -
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) ... -
Uniqueness and Complexity in Generalised Colouring
(University of Waterloo, 2003)The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition ...