Flexible List Colorings in Graphs with Special Degeneracy Conditions
Peter Bradshaw, Tom\'a\v{s} Masa\v{r}\'ik, Ladislav Stacho

TL;DR
This paper explores flexible list coloring in graphs with specific degeneracy properties, providing bounds and characterizations for various graph classes, and introduces a new concept of flexible degeneracy.
Contribution
It characterizes $rac{1}{6 ext{Δ}}$-flexible $ ext{Δ}$-choosability for graphs with maximum degree Δ and introduces flexible degeneracy, advancing understanding of list coloring with requests.
Findings
Graphs of maximum degree Δ (not isomorphic to K_{Δ+1}) are $rac{1}{6 ext{Δ}}$-flexibly Δ-choosable.
Graphs of treewidth 2 are $rac{1}{3}$-flexibly 3-choosable.
Graphs of treedepth k are $rac{1}{k}$-flexibly k-choosable.
Abstract
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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Nuclear Receptors and Signaling
