On graphs without four-vertex induced subgraphs

Given a family F of graphs, a graph G is F-free if it does not contain any graph in F as an induced subgraph. The problem of determining the complexity of colouring (claw, 4K1)- free graphs is a well-known open problem. In this paper we solve the colouring problem for a subclass of (claw, 4K1)-free graphs. We design a polynomial-time algorithm to colour (claw, 4K1, bridge, C4-twin)-free graphs. This algorithm is derived from a structural theorem on (claw, 4K1, bridge, C4-twin)-free graphs.