The L(2,1)-choosability of cycle

For a given graph $G=(V,E)$, let $mathscr L(G)={L(v) : vin V}$ be a prescribed list assignment. $G$ is $mathscr L$-$L(2,1)$-colorable if there exists a vertex labeling $f$ of $G$ such that $f(v)in L(v)$ for all $v in V$; $|f(u)-f(v)|geq 2$ if $d_G(u,v) = 1$; and $|f(u)-f(v)|geq 1$ if $d_G(u,v)=2$. If $G$ is $mathscr L$-$L(2,1)$-colorable for every list assignment $mathscr L$ with $|L(v)|geq k$ for all $vin V$, then $G$ is said to be $k$-$L(2,1)$-choosable. In this paper, we prove all cycles are $5$-$L(2,1)$-choosable.