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

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.