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DOI: 10.4230/LIPIcs.ISAAC.2020.31

Flexible List Colorings in Graphs with Special Degeneracy Conditions

Authors: Peter Bradshaw, Tomáš Masařk, and Ladislav Stacho

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract

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 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 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 Δ are flexibly (Δ - 1)-degenerate.

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Peter Bradshaw, Tomáš Masařk, and Ladislav Stacho. Flexible List Colorings in Graphs with Special Degeneracy Conditions. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bradshaw_et_al:LIPIcs.ISAAC.2020.31,
  author =	{Bradshaw, Peter and Masa\v{r}k, Tom\'{a}\v{s} and Stacho, Ladislav},
  title =	{{Flexible List Colorings in Graphs with Special Degeneracy Conditions}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{31:1--31:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.31},
  URN =		{urn:nbn:de:0030-drops-133750},
  doi =		{10.4230/LIPIcs.ISAAC.2020.31},
  annote =	{Keywords: Flexibility, List Coloring, Choosability, Degeneracy}
}

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Flexible List Colorings in Graphs with Special Degeneracy Conditions

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