Достаточные условия $2$-дистанционной $\Delta+1$ раскрашиваемости плоских графов

A trivial lower bound for the $2$-distance chromatic number $\chi_2(G)$ of any graph $G$ with maximum degree $\Delta$ is $\Delta+1$. We prove that if $G$ is planar and its girth is at least $7$, then $\chi_2(G)=\Delta+1$ whenever $\Delta\ge 30$. On the other hand, we construct planar graphs with girth $5$ and $6$ that have arbitrarily large $\Delta$ and $\chi_2(G)>\Delta+1$.