Equality cases for a bound on the chromatic number

It is known that the inequality $$\frac{\chi (G)(\chi (G)-1)}{2}+\mathrm{\mid }V\mathrm{\mid }-\chi (G)\le \mathrm{\mid }E\mathrm{\mid }$$ 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.