  It is known that the inequality $$ \frac{\chi(G)(\chi(G)-1)}{2} + |V| -
\chi(G) \leq |E|$$ holds for all connected graphs, where $\chi(G)$ denotes the
chromatic number of $G$. We prove that equality holds whenever the graph
consists of a complete graph or an odd cycle, together with finitely many trees
attached to its vertices.
