Extremal Graph Theory for Minors, Improper Colourings and Gonality

A graph, consists of a collection of vertices, some of which are joined by edges. Graphs are extremely useful for modelling and solving a wide range of real world problems, from traffic congestion to scheduling problems. We answer several questions about abstract graphs. We show that if a graph has five times as many edges as vertices, then it contains a structure known as a Petersen Minor, and present a formula for generating bounds of this form. We also find new results for variants of vertex colourings, and answer three recently posed questions about a chip firing game on graphs.