The list L(2,1)-labeling of planar graphs

Let $\mathbf{N}$ be the set of all positive integers. A list assignment of a graph $G$ is a function $L:V\left(G\right)?2^{\mathbf{N}}$ that assigns each vertex $v$ a list $L\left(v\right)$ for all $v\in V\left(G\right)$. We say that $G$ is $L$-$(2,1)$-choosable if there exists a function $?$ such that $?\left(v\right)\in L\left(v\right)$ for all $v\in V\left(G\right)$, $|?\left(u\right)??\left(v\right)|\ge 2$ if $u$ and $v$ are adjacent, and $|?\left(u\right)??\left(v\right)|\ge 1$ if $u$ and $v$ are at distance 2. The list-$L(2,1)$-labeling number ${\lambda }_l\left(G\right)$ of $G$ is the minimum $k$ such that for every list assignment $L=\{L\left(v\right):|L\left(v\right)|=k,v\in V\left(G\right)\}$, $G$ is $L$-$(2,1)$-choosable. We prove that if $G$ is a planar graph with girth $g\ge 8$ and its maximum degree $\Delta$ is large enough, then ${\lambda }_l\left(G\right)\le \Delta +3$. There are graphs with large enough $\Delta$ and $g\ge 8$ having ${\lambda }_l\left(G\right)=\Delta +3$.