Reconfiguration of Graph Colorings

Abstract

For this thesis, we consider two $k$-colorings of a graph $G$ adjacent if one can recolor one into the other by changing the color of one vertex. The reconfiguration graph of a graph $G$ on $k$ colors $\mathcal{C}_{k}(G)$ is the graph for which the vertices are the $k$-colorings of $G$, and an edge is between two $k$-colorings if they are adjacent. We further investigate the diameter of the reconfiguration graph on $k$ colors: $\mathrm{diam}\left(\mathcal{C}_{k}(G)\right)$. 
The general conjecture the thesis is based around says that $\mathrm{diam}\left(\mathcal{C}_{k}(G)\right)$ for every graph $G$ and $k \geq \Delta(G)+2$. This conjecture is confirmed for various families of graphs, for example the complete graph $K_n$ and complete bipartite $K_{n,m}$. This thesis will prove the lower bound of the conjecture for the family of complete $r$-partite graphs $G = K_{x_1,x_2,\ldots,x_r}$, utilising an approach from Cambie et al. for the proof. Furthermore we give an algorithm that computes the $k$-colorings of $G$, the reconfiguration graph $\mathcal{C}_{k}(G)$, and its diameter $\mathrm{diam}\left(\mathcal{C}_{k}(G)\right)$ and give a few results on this diameter for small graphs.