For a given ε>0, we say that a graph G is
ε-flexibly k-choosable if the following holds: for any assignment
L of color lists of size k on V(G), if a preferred color from a list is
requested at any set R of vertices, then at least ε∣R∣ of these
requests are satisfied by some L-coloring. We consider the question of
flexible choosability in several graph classes with certain degeneracy
conditions. We characterize the graphs of maximum degree Δ that are
ε-flexibly Δ-choosable for some ε=ε(Δ)>0, which answers a question of Dvo\v{r}\'ak, Norin, and
Postle [List coloring with requests, JGT 2019]. In particular, we show that for
any Δ≥3, any graph of maximum degree Δ that is not isomorphic
to KΔ+1 is 6Δ1-flexibly Δ-choosable. Our
fraction of 6Δ1 is within a constant factor of being the best
possible. We also show that graphs of treewidth 2 are 31-flexibly
3-choosable, answering a question of Choi et al.~[arXiv 2020], and we give
conditions for list assignments by which graphs of treewidth k are
k+11-flexibly (k+1)-choosable. We show furthermore that graphs of
treedepth k are k1-flexibly k-choosable. Finally, we introduce a
notion of flexible degeneracy, which strengthens flexible choosability, and we
show that apart from a well-understood class of exceptions, 3-connected
non-regular graphs of maximum degree Δ are flexibly (Δ−1)-degenerate.Comment: 21 pages, 5 figure