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

TL;DR

This paper investigates the computational complexity and extremal properties of a graph coloring problem involving partitioning vertices into independent and 2-independent sets, proving NP-Completeness and establishing tight bounds for various classes of planar graphs.

Contribution

It establishes NP-Completeness for the $(1^a, 2^b)$-coloring problem on planar graphs and provides tight extremal bounds for the number of 2-independent sets in specific graph classes.

Findings

NP-Completeness for certain planar graph classes

K-degenerate graphs are $(1^k, 2^{O( oot{n}{})})$-colorable

Triangle-free planar graphs are $(1^2, 2^{O( oot{n}{})})$-colorable

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.