Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Flexible List Colorings in Graphs with Special Degeneracy Conditions   For a given $\varepsilon > 0$, we say that a graph $G$ is
$\varepsilon$-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 $\varepsilon |R|$ 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 $\Delta$ that are
$\varepsilon$-flexibly $\Delta$-choosable for some $\varepsilon =
\varepsilon(\Delta) > 0$, 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 $\Delta\geq 3$, any graph of maximum degree $\Delta$ that is not isomorphic
to $K_{\Delta+1}$ is $\frac{1}{6\Delta}$-flexibly $\Delta$-choosable. Our
fraction of $\frac{1}{6 \Delta}$ is within a constant factor of being the best
possible. We also show that graphs of treewidth $2$ are $\frac{1}{3}$-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
$\frac{1}{k+1}$-flexibly $(k+1)$-choosable. We show furthermore that graphs of
treedepth $k$ are $\frac{1}{k}$-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 $\Delta$ are flexibly $(\Delta -
1)$-degenerate.

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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga