  For a colour cluster $\mathbb{C} =(\mathcal{C}_1,\mathcal{C}_2,
\mathcal{C}_3,\ldots,\mathcal{C}_\ell)$, where $\mathcal{C}_i$ is a colour
class such that $|\mathcal{C}_i|=r_i$, a positive integer, we investigate two
types of simple connected graph structures $G^{\mathbb{C}}_1$,
$G^{\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster
such that the chromatic numbers
$\chi(G^{\mathbb{C}}_1)=\chi(G^{\mathbb{C}}_2)=\ell$ and
$\min\{\varepsilon(G^{\mathbb{C}}_1)\}=\min\{\varepsilon(G^{\mathbb{C}}_2)\}
=\sum\limits_{i=1}^{\ell}r_i-1$. Therefore, the problem is the edge-minimality
inverse to finding the chromatic number of a given simple connected graph. In
this paper, we also discuss the chromatic Zagreb indices corresponding to
$G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$.
