Between proper and square coloring of planar graphs, hardness and extremal graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911446535766016 |
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| author | Delépine, Thomas |
| author_facet | Delépine, Thomas |
| contents | $(1^a, 2^b)$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13037 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Between proper and square coloring of planar graphs, hardness and extremal graphs Delépine, Thomas Combinatorics Computational Complexity $(1^a, 2^b)$-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. |
| title | Between proper and square coloring of planar graphs, hardness and extremal graphs |
| topic | Combinatorics Computational Complexity |
| url | https://arxiv.org/abs/2602.13037 |