Coloring squares of planar graphs with small maximum degree

For a graph $G$, by ${\chi }_2(G)$ we denote the minimum integer $k$, such that there is a $k$-coloring of the vertices of $G$ in which vertices at distance at most 2 receive distinct colors. Equivalently, ${\chi }_2(G)$ is the chromatic number of the square of $G$. In 1977 Wegner conjectured that if $G$ is planar and has maximum degree $\mathrm{\Delta }$, then ${\chi }_2(G)\le 7$ if $\mathrm{\Delta }\le 3$, ${\chi }_2(G)\le \mathrm{\Delta }+5$ if $4\le \mathrm{\Delta }\le 7$, and $\lfloor 3\Delta/2 \rfloor +1$if$\Delta \geq 8$. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of $\mathrm{\Delta }$. In this work we show that for every planar graph $G$ with maximum degree $\mathrm{\Delta }$ it holds that ${\chi }_2(G)\le 3\mathrm{\Delta }+4$. This result provides the best known upper bound for $6\le \mathrm{\Delta }\le 14$.