  Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$.
A proper $L$-list-coloring of $G$ is a function that maps every vertex to a
color in $L(v)$ such that no pair of adjacent vertices have the same color. We
say that a graph is list $k$-colorable when every vertex $v$ has a list of
colors of size at least $k$. A $2$-distance coloring is a coloring where
vertices at distance at most 2 cannot share the same color. We prove the
existence of a $2$-distance list ($\Delta+2$)-coloring for planar graphs with
girth at least $10$ and maximum degree $\Delta\geq 4$.
