In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let G be a connected graph with n vertices and maximum degree Δ(G). Let Rk(G) denote the graph with vertex set all proper k-colourings of G and two k-colourings are joined by an edge if they differ on the colour of exactly one vertex.
Our first main result states that RΔ(G)+1(G) has a unique non-trivial component with diameter O(n2). This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree.
A Kempe change is the operation of swapping some colours a, b of a component of the subgraph induced by vertices with colour a or b. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all Δ(G)-colourings of a graph G are Kempe equivalent unless G is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007).
Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs.
Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees