A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($\Delta+1$)-coloring for graphs with maximum average degree less than $\frac{18}{7}$ and maximum degree $\Delta\geq 7$. As a corollary, every planar graph with girth at least $9$ and $\Delta\geq 7$ admits a $2$-distance $(\Delta+1)$-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on $2$-distance coloring.

中文翻译：

---

使用电位方法对稀疏图进行$ 2 $距离$（Δ+ 1）$着色

图的$ 2 $距离$ k $着色是顶点的适当$ k $着色，其中距离最大为2的顶点不能共享相同的颜色。对于最大平均度小于$ \ frac {18} {7} $且最大度$ \ Delta \ geq 7 $的图，我们证明了存在$ 2 $距离（$ \ Delta + 1 $）着色。作为推论，每个周长至少为$ 9 $和$ \ Delta \ geq 7 $的平面图都接受$ 2 $距离的$（\ Delta + 1）$着色。与$ 2 $距离着色的经典方法相比，该证明使用了潜在的方法来减少新的配置。