It is known that the inequality $$ \frac{χ(G)(χ(G)-1)}{2} + |V| - χ(G) \leq |E|$$ holds for all connected graphs, where $χ(G)$ 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.