Robin Thomas asked whether for every proper minor-closed class $\mathcal{G}$, there exists a polynomial-time algorithm approximating the chromatic number of graphs from $\mathcal{G}$ up to a constant additive error independent on the class $\mathcal{G}$. We show this is not the case: unless $\text{P}=\text{NP}$, for every integer $k\ge 1$, there is no polynomial-time algorithm to color a $K_{4k+1}$-minor-free graph G using at most $\chi (G)+k-1$ colors. More generally, for every $k\ge 1$ and $1\le \beta \le 4/3$, there is no polynomial-time algorithm to color a $K_{4k+1}$-minor-free graph G using less than $\beta \chi (G)+(4-3\beta )k$ colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes.

Furthermore, we give somewhat weaker non-approximability bound for $K_{4k+1}$-minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

罗宾·托马斯（Robin Thomas）询问每门适当的未成年人封闭课程 $\mathcal{G}$，存在一种多项式时间算法，可以从中近似图的色数 $\mathcal{G}$ 取决于类别的最大恒定附加误差 $\mathcal{G}$。我们证明情况并非如此：除非$\text{P}=\text{NP}$，对于每个整数 $ķ\ge \mathrm{1个}$，没有多项式时间算法可以为 ${ķ}_{4ķ+\mathrm{1个}}$-次要图G最多使用$\chi （G）+ķ-\mathrm{1个}$颜色。更普遍地说，$ķ\ge \mathrm{1个}$ 和 $\mathrm{1个}\le \beta \le 4/3$，没有多项式时间算法可以为 ${ķ}_{4ķ+\mathrm{1个}}$-次要无图G使用小于$\beta \chi （G）+（4-3\beta ）ķ$颜色。据我们所知，这是关于适当的小封闭类中色数的第一个非平凡的非近似结果。

此外，我们给出了较弱的非近似约束 ${ķ}_{4ķ+\mathrm{1个}}$无小图，没有大小为4的小方组。在积极方面，我们提出加法逼近算法，该算法的误差取决于所禁止的未成年人的顶点数，以及一种在没有图的情况下附加误差为6的算法。 4个周期。