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 $\hat{\kappa }\left(G\right)=({\kappa }_0\left(G\right),{\kappa }_1\left(G\right),\dots ,{\kappa }_{\theta \left(G\right)-1}\left(G\right))$ and $\hat{\lambda }\left(G\right)=({\lambda }_0\left(G\right),{\lambda }_1\left(G\right),\dots ,{\lambda }_{\chi \left(G\right)-1}\left(G\right))$ where ${\kappa }_l\left(G\right)$ (respectively, ${\lambda }_k\left(G\right)$) is the minimum $k$ (respectively, $l$) such that $G$ has a $(k,l)$-colouring. We prove that $\hat{\kappa }\left(G\right)$ and $\hat{\lambda }\left(G\right)$ 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 $\hat{\kappa }\left(G\right)$ or in $\hat{\lambda }\left(G\right)$. Using the decomposition property of cographs we show that every cograph can be represented by Ferrers diagram. We devise algorithms which compute $\hat{\kappa }\left(G\right)$ for cographs $G$ and find an induced subgraph in $G$ that can be used to certify the non-$(k,l)$-colourability of $G$.

对于一对自然数 $ķ，升$， 一种 $（ķ，升）$图的着色 $G$ 是顶点集的一部分 $G$ 成（可能为空）集 ${\mathrm{小号}}_{\mathrm{1个}}，{\mathrm{小号}}_2，\dots ，{\mathrm{小号}}_{ķ}$， $C_{\mathrm{1个}}，C_2，\dots ，C_{升}$ 这样每套 ${\mathrm{小号}}_{\mathrm{一世}}$ 是一个独立的集合，每个集合 $C_{Ĵ}$ 引起集团 $G$。的$（ķ，升）$对于特殊的图类，例如弦图，cograph和线图，已经研究了通常为NP完全的着色问题。让$\hat{\kappa }（G）=（{\kappa }_0（G），{\kappa }_{\mathrm{1个}}（G），\dots ，{\kappa }_{\theta （G）-\mathrm{1个}}（G））$ 和 $\hat{\lambda }（G）=（{\lambda }_0（G），{\lambda }_{\mathrm{1个}}（G），\dots ，{\lambda }_{\chi （G）-\mathrm{1个}}（G））$ 哪里 ${\kappa }_{升}（G）$ （分别， ${\lambda }_{ķ}（G）$）是最小值 $ķ$ （分别， $升$）这样 $G$ 有一个 $（ķ，升）$-染色。我们证明$\hat{\kappa }（G）$ 和 $\hat{\lambda }（G）$ 是每个图的一对共轭序列 $G$ 什么时候 $G$ 是一个cograph，其中的顶点数 $G$ 等于中的条目之和 $\hat{\kappa }（G）$ 或在 $\hat{\lambda }（G）$。利用字形图的分解特性，我们可以证明每个字形都可以用Ferrers图表示。我们设计算法$\hat{\kappa }（G）$ 用于合作伙伴 $G$ 并在中找到一个诱导子图 $G$ 可以用来证明非$（ķ，升）$的可着色性 $G$。