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 lists of size k on V (G), if a preferred color is requested at any set R of vertices, then at least ε|R| of these requests are satisfied by some L-coloring. We consider flexible list colorings in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree ∆ that are ε-flexibly ∆-choosable for some ε = ε(∆) > 0, which answers a question of Dvořák, Norin, and Postle [List coloring with requests, JGT 2019]. We also show that graphs of treewidth 2 are 1 3 -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 1 k+1 -flexibly (k + 1)-choosable. We show furthermore that graphs of treedepth k are 1 k -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 ∆ are flexibly (∆ − 1)-degenerate.

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A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geqslant 4$, every balanced bipartite graph on $2n$ vertices in which each vertex in one color class has degree greater than $\frac{n}{2}$ and each vertex in the other color class has degree at least $\frac{n}{2}$ is bipancyclic. We prove a generalization of this theorem in the setting of graph transversals. Namely, we show that given a family $\mathcal{G}$ of $2n$ bipartite graphs on a common set $X$ of $2n$ vertices with a common balanced bipartition, if each graph of $\mathcal G$ has minimum degree greater than $\frac{n}{2}$ in one color class and minimum degree at least $\frac{n}{2}$ in the other color class, then there exists a cycle on $X$ of each even length $4 \leqslant \ell \leqslant 2n$ that uses at most one edge from each graph of $\mathcal G$. We also show that given a family $\mathcal G$ of $n$ bipartite graphs on a common set $X$ of $2n$ vertices meeting the same degree conditions, there exists a perfect matching on $X$ that uses exactly one edge from each graph of $\mathcal G$.