Colouring diamond-free graphs

•We classify the complexity of Colouring for (diamond, H)-free graphs when |V(H)|≤5.•We generalize a known decomposition of bipartite graphs to k-partite graphs.•We find five new classes of (H1,H2)-free graphs of bounded clique-width.•This reduces the number of open cases from 13 to 8.

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 (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P1+2P2)-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 (H1,H2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H1,H2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.