An evaluation on the gracefulness and colouring of graphs

In this thesis we shall introduce two interesting topics from graph theory and begin to explore what happens when we combine these together. We will be focusing on an area known as graph colouring and assessing it alongside a very unique set of graphs called graceful graphs. The two topic areas, although not mixed together often, nicely support each other in introducing various findings from each of the topics. We will start by investigating graceful graphs and determining what classes of graph can be deemed to be graceful, before introducing some of the fundamentals of graph colouring. Following this we can then begin to investigate the two topics combined and will see a whole range of results, including some fascinating less well known discoveries. Furthermore, we will introduce some different types of graph colouring based off the properties of graceful graphs. Later in the thesis there will also be a focus on tree graphs, as they have had a huge influence on research involving graceful graphs over the years. We will then conclude by investigating some results that have been formulated by combining graceful graphs with a type of graph colouring known as total colouring.