Title:

Graph colouring with input restrictions

In this thesis, we research the computational complexity of the graph colouring problem and its variants including precolouring extension and list colouring for graph classes that can be characterised by forbidding one or more induced subgraphs. We investigate the structural properties of such graph classes and prove a number of new properties. We then consider to what extent these properties can be used for efficiently solving the three types of colouring problems on these graph classes. In some cases we obtain polynomialtime algorithms, whereas other cases turn out to be NPcomplete. We determine the computational complexity of kCOLOURING, kPRECOLOURING EXTENSION and LIST kCOLOURING on $P_k$free graphs. In particular, we prove that kCOLOURING on $P_8$free graphs is NPcomplete, 4PRECOLOURING EXTENSION $P_7$free graphs is NPcomplete, and LIST 4COLOURING on $P_6$free graphs is NPcomplete. In addition, we show the existence of an integer r such that kCOLOURING is NPcomplete for $P_r$free graphs with girth 4. In contrast, we determine for any fixed girth $g\geq 4$ a lower bound $r(g)$ such that every $P_{r(g)}$free graph with girth at least $g$ is 3colourable. We also prove that 3LIST COLOURING is NPcomplete for complete graphs minus a matching. We present a polynomialtime algorithm for solving 4PRECOLOURING EXTENSION on $(P_2+P_3)$free graphs, a polynomialtime algorithm for solving LIST 3Colouring on $(P_2+P_4)$free graphs, and a polynomialtime algorithm for solving LIST 3COLOURING on $sP_3$free graphs. We prove that LIST kCOLOURING for $(K_{s,t},P_r)$free graphs is also polynomialtime solvable. We obtain several new dichotomies by combining the above results with some known results.
