Flexible list colorings in graphs with special degeneracy conditions

For a given $\varepsilon > 0$, we say that graph $G$ is $\varepsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of color lists size $k$ on $V(G)$, preferred from list requested at set $R$ vertices, then least |R|$ these requests are satisfied by some $L$-coloring. We consider question flexible choosability in several classes with certain degeneracy conditions. characterize graphs maximum degree $\Delta$ $\Delta$-choosable = \varepsilon(\Delta) which answers Dvo\v{r}\'ak, Norin, and Postle [List coloring requests, JGT 2019]. In particular, show $\Delta\geq 3$, not isomorphic to $K_{\Delta+1}$ $\frac{1}{6\Delta}$-flexibly $\Delta$-choosable. Our fraction $\frac{1}{6 \Delta}$ within constant factor being best possible. also treewidth $2$ $\frac{1}{3}$-flexibly $3$-choosable, answering Choi et al.~[arXiv 2020], give conditions assignments $\frac{1}{k+1}$-flexibly $(k+1)$-choosable. furthermore treedepth $\frac{1}{k}$-flexibly $k$-choosable. Finally, introduce notion degeneracy, strengthens choosability, apart well-understood class exceptions, 3-connected non-regular flexibly $(\Delta - 1)$-degenerate.