Now showing items 1-4 of 4
The Erdős Pentagon Problem
(University of Waterloo, 2018-12-20)
The Erdős pentagon problem asks about the maximum number of copies of C_5 that one can find in a triangle-free graph. This problem was posed in 1984, but was not resolved until 2012. In this thesis, we aim to capture the ...
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: ...
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 ...
On Geometric Drawings of Graphs
(University of Waterloo, 2018-04-18)
This thesis is about geometric drawings of graphs and their topological generalizations. First, we study pseudolinear drawings of graphs in the plane. A pseudolinear drawing is one in which every edge can be extended ...