1 research outputs found
Flexible List Colorings in Graphs with Special Degeneracy Conditions
For a given , we say that a graph is
-flexibly -choosable if the following holds: for any assignment
of color lists of size on , if a preferred color from a list is
requested at any set of vertices, then at least of these
requests are satisfied by some -coloring. We consider the question of
flexible choosability in several graph classes with certain degeneracy
conditions. We characterize the graphs of maximum degree that are
-flexibly -choosable for some , 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 , any graph of maximum degree that is not isomorphic
to is -flexibly -choosable. Our
fraction of is within a constant factor of being the best
possible. We also show that graphs of treewidth are -flexibly
-choosable, answering a question of Choi et al.~[arXiv 2020], and we give
conditions for list assignments by which graphs of treewidth are
-flexibly -choosable. We show furthermore that graphs of
treedepth are -flexibly -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 -degenerate.Comment: 21 pages, 5 figure