  The Grundy number of a graph is the maximum number of colours used by the
"First-Fit" greedy colouring algorithm over all vertex orderings. Given a
vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring
algorithm colours the vertices in the order of $\sigma$ by assigning to each
vertex the smallest colour unused in its neighbourhood.
  By restricting this procedure to vertex orderings that are connected, we
obtain {\em connected greedy colourings}. For some graphs, all connected greedy
colourings use exactly $\chi(G)$ colours; they are called {\em good graphs}. On
the opposite, some graphs do not admit any connected greedy colouring using
only $\chi(G)$ colours; they are called {\em ugly graphs}.
  We show that no perfect graph is ugly. We also give simple proofs of this
fact for subclasses of perfect graphs (block graphs, comparability graphs), and
show that no $K_4$-minor free graph is ugly.
  Moreover, our proofs are constructive, and imply the existence of
polynomial-time algorithms to compute good connected orderings for these graph
classes.
