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 $|l(v)|geq k$ ...