论文标题
（k; l） - 彩色和弗雷尔图表的表达式
(k; l)-Colourings and Ferrers Diagram Representations of Cographs
论文作者
论文摘要
对于一对自然数量$ k，l $，a $（k，l）$ - 颜色的颜色$ g $是$ g $的$ g $的分区（可能为空的）集合$ s_1，s_2，s_k $，s_k $，s_k $，$ c_1，$ c_1，c_2，c_2，c_2，c_ dots，c_l $ s $ s $ s $ s_i $ s_i $ s_i $ s_i $ s $ s_i $ s_i $ s $ s $ s_i $ g $。 $（k，l）$ - 着色问题一般是NP完整的，已在特殊的图形类别（例如弦图，cographs和line Graphs）中进行了研究。令$ \hatκ（g）=（κ_0（g），κ_1（g），\ dots，κ__{θ（g）-1} -1}（g））$和$ \hatλ（g）=（λ_0（g），λ_1（G），λ_1（g），\ dots，\ dots，\ dots，\ dots，λ___________________$ where（g）$（g）$（G） - $λ_k（g）$）是最低$ k $（分别为$ l $），因此$ g $具有$（k，l）$ - 着色。我们证明，$ \hatκ（g）$和$ \hatλ$是每个图$ g $的一对共轭序列，而当$ g $是cograph时，$ g $中的顶点数量等于$ \hatκ（g）$或$ \hatλ（g）$中的$ \hatκ（g）$中的条目总和。使用Cographs的分解属性，我们表明每个Cograph可以用Ferrers图表示。我们设计了Cographs $ g $计算$ \hatκ（g）$的算法，并在$ g $中找到一个诱导的子图，可用于证明$ g $的非 - $（k，k，l）$ - 可着色性。
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 $\hatκ(G) = (κ_0(G),κ_1(G),\dots,κ_{θ(G)-1}(G))$ and $\hatλ(G) = (λ_0(G),λ_1(G),\dots,λ_{χ(G)-1}(G))$ where $κ_l(G)$ (respectively, $λ_k(G)$) is the minimum $k$ (respectively, $l$) such that $G$ has a $(k,l)$-colouring. We prove that $\hatκ(G)$ and $\hatλ(G)$ 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 $\hatκ(G)$ or in $\hatλ(G)$. Using the decomposition property of cographs we show that every cograph can be represented by Ferrers diagram. We devise algorithms which compute $\hatκ(G)$ for cographs $G$ and find an induced subgraph in $G$ that can be used to certify the non-$(k,l)$-colourability of $G$.
