  An ordering of the vertices of a graph is \emph{connected} if every vertex
(but the first) has a neighbor among its predecessors. The greedy colouring
algorithm of a graph with a connected order consists in taking the vertices in
order, and assigning to each vertex the smallest available colour. A graph is
\emph{good} if the greedy algorithm on every connected order gives every
connected induced subgraph of it an optimal colouring. We give the
characterization of good claw-free graphs in terms of minimal forbidden induced
subgraphs.
