Between proper and square coloring of planar graphs, hardness and extremal graphs

Abstract

$(1^a, 2^b)$-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.

Figures (11)

• Figure 1: The vertex gadget used in the proof of Theorem \ref{['thm:npc_1_2_coloring_for_every_g']}.
• Figure 2: Two graphs used in the proof of Theorem \ref{['thm:1_k_npc']}, $\textrm{VAR}_v^k$ is $(1^{1}, 2^{k})$-colorable and $v$ is colored with a distance-1 color if and only if $\bar{v}$ is colored with a distance-1 color, and $\textrm{CL}_C^k$ is $(1^{1}, 2^{k})$-colorable and at least one of $v_C^x$, $v_C^y$ or $v_C^z$ must be colored with a distance-2 color.
• Figure 3: The variable and clause gadgets used in the proof of Theorem \ref{['thm:2k_structural_gap']}.
• Figure 4: Two graphs constructed from copies of $G' - v$ and used in the proof of Proposition \ref{['prop:2_k_npc_tool']}.
• Figure 5: Vertex gadget used to represent a vertex $v$ in the proof of Theorem \ref{['thm:npc_gap_3_1']}.

Theorems & Definitions (54)

• Conjecture 1: Wegner wegner1977graphs, 1977
• Conjecture 2: Wang and Lih Wang_and_Lih, 2003
• Theorem 3: Choi and Liu choi2025propersquarecoloringssparse, 2025
• Theorem 4
• Theorem 5
• proof
• Theorem 6
• proof
• Lemma 7
• proof