Colouring Diamond-free Graphs

The Colouring problem is that of deciding, given a graph $G$ and an integer $k$, whether $G$ admits a (proper) $k$-colouring. For all graphs $H$ up to five vertices, we classify the computational complexity of Colouring for $(\mbox{diamond},H)$-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for $(\mbox{diamond}, P_1+2P_2)$-free graphs. Our technique for handling this case is to reduce the graph under consideration to a $k$-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of $(H_1,H_2)$-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of $(H_1,H_2)$-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.