图的完美性和$ω$ -可染性的禁导对
Forbidden induced pairs for perfectness and $ω$-colourability of graphs
摘要：
我们对图对$\{ X, Y \}$进行优化，使得所有无$\{ X, Y \}$的图（不同于$C_5$）都是完美的.类似地，我们对$\{ X, Y \}$进行双线性化，使得所有无$\{ X, Y \}$的图（不同于$C_5$）都是$ω$ -簇（即它们的色数等于它们的团数）。更一般地说，我们展示了对$\{ X, Y \}$的完美性和$ω$ -着色的所有连接$\{ X, Y \}$ -免费的图形是独立的至少$3$，不同于一个奇数周期，并至少$n_0$，和类似的刻画受到这些额外的约束的每个子集的特征。(The类是非遗传的，并且完美性和$ω$可着色性的特征是不同的。）我们建立在Brause等人的最近结果上。关于$\{ K_{1,3}, Y \}$ -free图，我们使用Ramsey定理和强完美图定理作为主要工具。我们目前的特征，已知的结果禁止对$χ$ -有界性和决定$k$ -着色在多项式时间。
英文摘要：
We characterise the pairs of graphs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are perfect. Similarly, we characterise pairs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are $ω$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $ω$-colourability of all connected $\{ X, Y \}$-free graphs which are of independence at least $3$, distinct from an odd cycle, and of order at least $n_0$, and similar characterisations subject to each subset of these additional constraints. (The classes are non-hereditary and the characterisations for perfectness and $ω$-colourability are different.) We build on recent results of Brause et al. on $\{ K_{1,3}, Y \}$-free graphs, and we use Ramsey's Theorem and the Strong Perfect Graph Theorem as main tools. We relate the present characterisations to known results on forbidden pairs for $χ$-boundedness and deciding $k$-colourability in polynomial time.
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