  This work introduces the concept of \emph{upper-critical graphs}, in a
complementary way of the conventional (lower)critical graphs: an element $x$ of
a graph $G$ is called \emph{critical} if $\chi(G-x)<\chi(G)$. It is said that
$G$ is a \emph{critical graph} if every element (vertex or edge) of $G$ is
critical. Analogously, a graph $G$ is called \emph{upper-critical} if there is
no edge that can be added to $G$ such that $G$ preserves its chromatic number,
i.e. \{$e \in E(\bar{G}) \; | \; \chi(G+e) = \chi(G)$ \} $=$ $\emptyset$. We
show that the class of upper-critical graphs is the same as the class of
complete $k$-partite graphs. A characterization in terms of hereditary
properties under some transformations, e.g. subgraphs and minors and in terms
of construction and counting is given.
