Flexible List Colorings in Graphs with Special Degeneracy Conditions
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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 may be satisfied by some L-coloring. We consider flexible list colorings in several graph classes with certain special 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 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 I. 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, three-connected non-regular graphs of maximum degree Δ are flexibly (Δ - 1)-degenerate.
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