# Flexibility of planar graphs of girth at least six

@article{Dvok2020FlexibilityOP, title={Flexibility of planar graphs of girth at least six}, author={Z. Dvoř{\'a}k and Tom{\'a}{\vs} Masař{\'i}k and Jan Mus{\'i}lek and Ondrej Pangr{\'a}c}, journal={ArXiv}, year={2020}, volume={abs/1902.04069} }

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at least a constant fraction of the preferences.

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#### 6 Citations

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- Mathematics
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Let $G$ be a $\{C_4, C_5\}$-free planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. We prove that if all lists have size at least four, then there… Expand

Flexibility of planar graphs without 4-cycles

- Mathematics, Computer Science
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It is proved that if G is a planar graph without 4-cycles and all lists have size at least five, then there exists an L-coloring respecting at least a constant fraction of the preferences. Expand

Flexibility of planar graphs without 4-cycles

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It is proved that if G is a planar graph without 4-cycles and all lists have size at least five, then there exists an L-coloring respecting at least a constant fraction of the preferences. Expand

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The notion of weak flexibility is introduced, where $R = V, by showing that for every positive integer $b$ there exists $\epsilon(b)>0$ so that the class of planar graphs without $K_4, C-5, C_6 , C_7, B_5$ is weakly $\ep silon-flexible for lists of size $4". Expand

Flexibility of Planar Graphs - Sharpening the Tools to Get Lists of Size Four

- Computer Science, Mathematics
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A variant of the widely studied class of precoloring extension problems from [Z. Dvořak, S. Norin, and L. Postle]: List coloring with requests is looked at and a stronger version of the main tool used in the proofs of the aforementioned results is given. Expand

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The notion of flexible degeneracy is introduced, which strengthens flexible choosability, and it is shown that apart from a well-understood class of exceptions, three-connected non-regular graphs of maximum degree $\Delta$ are flexibly $(\Delta - 1)$-degenerate. Expand

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