As the class $\mathcal T_4$ of graphs of twin-width at most 4 contains every finite subgraph of the infinite grid and every graph obtained by subdividing each edge of an $n$-vertex graph at least $2 \log n$ times, most NP-hard graph problems, like Max Independent Set, Dominating Set, Hamiltonian Cycle, remain so on $\mathcal T_4$. However, Min Coloring and k-Coloring are easy on both families because they are 2-colorable and 3-colorable, respectively. We show that Min Coloring is NP-hard on the class $\mathcal T_3$ of graphs of twin-width at most 3. This is the first hardness result on $\mathcal T_3$ for a problem that is easy on cographs (twin-width 0), on trees (whose twin-width is at most 2), and on unit circular-arc graphs (whose twin-width is at most 3). We also show that for every $k \geqslant 3$, k-Coloring is NP-hard on $\mathcal T_4$. We finally make two observations: (1) there are currently very few problems known to be in P on $\mathcal T_d$ (graphs of twin-width at most $d$) and NP-hard on $\mathcal T_{d+1}$ for some nonnegative integer $d$, and (2) unlike $\mathcal T_4$, which contains every graph as an induced minor, the class $\mathcal T_3$ excludes a fixed planar graph as an induced minor; thus it may be viewed as a special case (or potential counterexample) for conjectures about classes excluding a (planar) induced minor. These observations are accompanied by several open questions.

翻译：由于双宽至多为4的图类$\mathcal T_4$包含无限网格的每个有限子图，以及通过对$n$顶点图的每条边至少细分$2 \log n$次得到的每个图，大多数NP难图问题（如最大独立集、支配集、哈密顿环）在$\mathcal T_4$上仍保持NP难性。然而，最小着色问题和k-着色问题在这两类图上都是容易的，因为它们分别是2-可着色和3-可着色的。我们证明最小着色问题在双宽至多为3的图类$\mathcal T_3$上是NP难的。这是针对在余图（双宽0）、树（双宽至多为2）和单位圆弧图（双宽至多为3）上均为易解的问题，在$\mathcal T_3$上获得的首个困难性结果。我们还证明对于每个$k \geqslant 3$，k-着色问题在$\mathcal T_4$上是NP难的。最后我们提出两点观察：(1)目前已知在$\mathcal T_d$（双宽至多为$d$的图）上属于P类，而在某个非负整数$d$对应的$\mathcal T_{d+1}$上为NP难的问题非常少；(2)与包含每个图作为诱导子式的$\mathcal T_4$不同，图类$\mathcal T_3$排除某个固定平面图作为诱导子式；因此可将其视为关于排除（平面）诱导子式的图类猜想的特例（或潜在反例）。这些观察伴随着若干开放性问题。