Between proper and square colorings of planar graphs with maximum degree at most four

TL;DR

This paper investigates intermediate coloring schemes for planar graphs with maximum degree four, proving new bounds on packing colorings that relate to classical proper and square colorings.

Contribution

It introduces and proves two new packing coloring bounds for planar graphs with maximum degree four, advancing understanding of intermediate coloring schemes.

Findings

Every such graph is packing (1,2^{10})-colorable.

Every such graph is packing (1^2,2^7)-colorable.

Results relate to and extend classical coloring conjectures.

Abstract

An $i$-independent set is a vertex set whose pairwise distance is at least $i+1$. A proper (square) $k$-coloring of a graph $G$ is a partition of its vertex set into $k$ independent ($2$-independent) sets. A packing $(1^{j}, 2^k)$-coloring of a graph $G$ is a partition of $V(G)$ into $j$ independent sets and $k$ $2$-independent sets. It can be viewed as intermediate colorings between proper and square coloring. Wegner conjectured in 1977 that every planar graph with maximum degree at most four is square $9$-colorable. Bousquet, Deschamps, de Meyer, and Pierron proved an upper bound of $12$, which is the current best result toward the conjecture of Wegner. In this paper, we prove two analogue results that every planar graph with maximum degree at most four is packing $(1,2^{10})$-colorable and packing $(1^2,2^7)$-colorable.