(k; l)-Colourings and Ferrers Diagram Representations of Cographs

For a pair of natural numbers $k, l$, a $(k,l)$-colouring of a graph $G$ is a partition of the vertex set of $G$ into (possibly empty) sets $S_1, S_2, \dots, S_k$, $C_1, C_2, \dots, C_l$ such that each set $S_i$ is an independent set and each set $C_j$ induces a clique in $G$. The $(k,l)$-colouring problem, which is NP-complete in general, has been studied for special graph classes such as chordal graphs, cographs and line graphs. Let $\hat{\kappa}(G) = (\kappa_0(G),\kappa_1(G),\dots,\kappa_{\theta(G)-1}(G))$ and $\hat{\lambda}(G) = (\lambda_0(G),\lambda_1(G),\dots,\lambda_{\chi(G)-1}(G))$ where $\kappa_l(G)$ (respectively, $\lambda_k(G)$) is the minimum $k$ (respectively, $l$) such that $G$ has a $(k,l)$-colouring. We prove that $\hat{\kappa}(G)$ and $\hat{\lambda}(G)$ are a pair of conjugate sequences for every graph $G$ and when $G$ is a cograph, the number of vertices in $G$ is equal to the sum of the entries in $\hat{\kappa}(G)$ or in $\hat{\lambda}(G)$. Using the decomposition property of cographs we show that every cograph can be represented by Ferrers diagram. We devise algorithms which compute $\hat{\kappa}(G)$ for cographs $G$ and find an induced subgraph in $G$ that can be used to certify the non-$(k,l)$-colourability of $G$.