  A proper vertex colouring of a graph is \emph{nested} if the vertices of each
of its colour classes can be ordered by inclusion of their open neighbourhoods.
Through a relation to partially ordered sets, we show that the nested chromatic
number can be computed in polynomial time.
  Clearly, the nested chromatic number is an upper bound for the chromatic
number of a graph. We develop multiple distinct bounds on the nested chromatic
number using common properties of graphs. We also determine the behaviour of
the nested chromatic number under several graph operations, including the
direct, Cartesian, strong, and lexicographic product. Moreover, we classify
precisely the possible nested chromatic numbers of graphs on a fixed number of
vertices with a fixed chromatic number.
