Frugal coloring in planar graphs with maximum degree 4 and 5

Source abstract

A proper coloring of vertices of a graph $G$ is called $t$-frugal if no color appears more than $t$ times in the neighborhood of any vertex of $G$. While $1$-frugal coloring coincides with $2$-distance coloring, we study $2$-frugal colorings of planar graphs with maximum degree $4$ and $5$, respectively. We prove that every planar graph with maximum degree $4$ admits a $2$-frugal coloring with $7$ colors, while a planar graph with maximum degree $5$ admits a $2$-frugal coloring with $10$ colors. The results present variations of parts of the famous Wegner's conjecture, which for planar graphs with maximum degree $Δ\in\{4,\ldots,7\}$ states that their $2$-distance chromatic number is at most $Δ+5$.