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 $k$-coloring of vertices $G$ in which at distance most 2 receive distinct colors. Equivalently, chromatic number square $G$. In 1977 Wegner conjectured if planar and has maximum degree $\Delta$, then $\chi_2(G) \leq 7$ $\Delta 3$, \Delta+5$ $4 \Delta 7$, $\lfloor 3\Delta/2 \rfloor +1$ \geq 8$. Despite extensive work, known upper bounds are quite far from ones, especially for small values $\Delta$. this work show every with $\Delta$ it holds 3\Delta+4$. This result provides best bound $6 14$.