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We study a family $\mathcal{B}$ of pseudo-multipartite graphs indexed bystaircase partitions. They are realised from the reduced words of certain classof permutations. We investigate the vertex proper colouring of these graphs andgive the general chromatic polynomial. For each member $B_{\lambda}$, weconstruct an affine toric ideal $\mathcal{I}_{B_{\lambda}}$ associated tovertex proper colouring using partition identity. It turns out that theprojective version $\mathcal{V}(\mathcal{I}_{B_{\lambda}})$ is realised fromthe cartoon diagram associated with the vertex proper colouring.

On The Toric Ideals of the Coloured Graphs of Reduced Words