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 $G^{ \mathbb{ C}}_1$, $G^{ \mathbb{ C}}_2$.
Categories: math.GM