$(1^a, 2^b)$-coloring is the problem of partitioning the vertex set of a graph into $a$ independent sets and $b$ 2-independent sets. This problem was recently introduced by Choi and Liu. We study the computational complexity and extremal properties of $(1^a, 2^b)$-coloring. We prove that this problem is NP-Complete even when restricted to certain classes of planar graphs, and we also investigate the extremal values of $b$ when $a$ is fixed and in some $(a + 1)$-colorable classes of graphs. In particular, we prove that $k$-degenerate graphs are $(1^k, 2^{O(\sqrt{n})})$-colorable, that triangle-free planar graphs are $(1^2, 2^{O(\sqrt{n})})$-colorable and that planar graphs are $(1^3, 2^{O(\sqrt{n})})$-colorable. All upper bounds obtained are tight up to a constant factor.

翻译：$(1^a, 2^b)$-着色问题是将图的顶点集划分为 $a$ 个独立集与 $b$ 个 2-独立集的问题。该问题由 Choi 与 Liu 近期提出。本文研究了 $(1^a, 2^b)$-着色的计算复杂性及极值性质。我们证明了即使限制在特定的平面图类上，该问题仍然是 NP-完全的，并进一步探究了当 $a$ 固定时 $b$ 的极值，以及在某些 $(a + 1)$-可着色图类中的情况。具体而言，我们证明了 $k$-退化图是 $(1^k, 2^{O(\sqrt{n})})$-可着色的，无三角形平面图是 $(1^2, 2^{O(\sqrt{n})})$-可着色的，且平面图是 $(1^3, 2^{O(\sqrt{n})})$-可着色的。所获得的所有上界在常数因子范围内均是紧的。