The NP-complete problems Colouring and k-Colouring $(k\geq 3$) are well studied on $H$-free graphs, i.e., graphs that do not contain some fixed graph $H$ as an induced subgraph. We research to what extent the known polynomial-time algorithms for $H$-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. For a graph $H$, a partitioned probe $H$-free graph $(G,P,N)$ consists of a graph $G=(V,E)$, together with a set $P\subseteq V$ of probes and an independent set $N=V\setminus P$ of non-probes, such that $G+F$ is $H$-free for some edge set $F\subseteq \binom{N}{2}$. We first fully classify the complexity of Colouring on partitioned probe $H$-free graphs and show that this dichotomy is different from the known dichotomy of Colouring for $H$-free graphs. Our main result is a dichotomy of $3$-Colouring for partitioned probe $P_t$-free graphs: we prove that the problem is polynomial-time solvable if $t\leq 5$ but NP-complete if $t\geq 6$. In contrast, $3$-Colouring on $P_t$-free graphs is known to be polynomial-time solvable if $t\leq 7$ and quasi polynomial-time solvable for $t\geq 8$.

翻译：NP完全问题着色和k-着色$(k\geq 3$)在$H$-自由图（即不包含固定图$H$作为诱导子图的图）上的研究已较为深入。我们探讨：若仅已知输入图的部分边，现有针对$H$-自由图的多项式时间算法可在多大程度上推广。为此，我们采用九十年代初提出的经典探针图模型进行研究。对于图$H$，划分式探针$H$-自由图$(G,P,N)$由以下部分组成：图$G=(V,E)$、探针顶点集$P\subseteq V$以及独立非探针顶点集$N=V\setminus P$，且存在边集$F\subseteq \binom{N}{2}$使得$G+F$为$H$-自由图。我们首先完整分类了划分式探针$H$-自由图上着色问题的复杂度，并证明该二分性质与已知的$H$-自由图着色二分性质不同。我们的主要成果是划分式探针$P_t$-自由图上3-着色问题的二分性证明：当$t\leq 5$时问题可在多项式时间内求解，而当$t\geq 6$时问题为NP完全。相比之下，已知$P_t$-自由图上的3-着色问题在$t\leq 7$时存在多项式时间算法，在$t\geq 8$时存在拟多项式时间算法。