 Clique-independent partition class

Graphs in $\mathcal{C}(a,b)$ admit a partition into $a$ total classes, of which $b$ are designated cliques and $a-b$ are designated independent sets. Cross edges are arbitrary and empty classes are allowed. Balanced cross-edge choices give quadratic labelled speed coefficient $1-1/a$. Comparing this count with every $(a+1)$-class type shows that the hereditary colouring number of this class is exactly $a$.