# 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.
6 Citations

#### Topics from this paper

Flexibility of planar graphs without $C_4$ and $C_5$
• Mathematics
• 2020
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 thereExpand
Flexibility of planar graphs without 4-cycles
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
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
On Weak Flexibility in Planar Graphs
• Mathematics, Computer Science
• ArXiv
• 2020
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 $4". Expand Flexibility of Planar Graphs - Sharpening the Tools to Get Lists of Size Four • Computer Science, Mathematics • ArXiv • 2020 A variant of the widely studied class of precoloring extension problems from [Z. Dvořak, S. Norin, and L. Postle]: List coloring with requests is looked at and a stronger version of the main tool used in the proofs of the aforementioned results is given. Expand Flexible List Colorings in Graphs with Special Degeneracy Conditions • Mathematics, Computer Science • ISAAC • 2020 The notion of flexible degeneracy is introduced, which strengthens flexible choosability, and it is shown that apart from a well-understood class of exceptions, three-connected non-regular graphs of maximum degree$\Delta$are flexibly$(\Delta - 1)$-degenerate. Expand #### References SHOWING 1-10 OF 11 REFERENCES Flexibility of triangle-free planar graphs • Mathematics, Computer Science • J. Graph Theory • 2021 It is proved that if G is triangle-free and all lists have size at least four, then there exists an L-coloring respecting at least a constant fraction of the preferences. Expand 3-List-Coloring Planar Graphs of Girth 5 • C. Thomassen • Computer Science, Mathematics • J. Comb. Theory, Ser. B • 1995 Abstract We prove that every planar graph of girth at least 5 is 3-choosable. It is even possible to precolor any 5-cycle in the graph. This extension implies Grotzsch′s theorem that every planarExpand List coloring with requests • Mathematics, Computer Science • J. Graph Theory • 2019 Several natural questions arising in this context of L-coloring G are explored, and directions for further research are proposed. Expand Color-Critical Graphs on a Fixed Surface • C. Thomassen • Computer Science, Mathematics • J. Comb. Theory, Ser. B • 1997 A polynomially bounded algorithm for deciding if a graph onScan is bek-colored is extended to the case where a subgraph of fixed cardinality is precolored and a corresponding list-color theorem is established. Expand A not 3-choosable planar graph without 3-cycles • M. Voigt • Computer Science, Mathematics • Discret. Math. • 1995 The question resulted whether every planar graph without 3-cycles is 3-choosable is given, and it is proved that everyPlanar graph with girth greater than 4 is 3 - Choosable. Expand Do Triangle-Free Planar Graphs have Exponentially Many$3$-Colorings? • Mathematics, Computer Science • Electron. J. Comb. • 2017 The conjecture that triangle-free planar graphs have an exponential number of$3$-colorings is shown to be equivalent to the following statement: there exists a positive real$\alpha$such that$G-(A\setminus A')$is$3-colorable. Expand
Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces
• L. Postle
• Mathematics, Computer Science
• 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
• 2019
The first linear-time algorithms to find fixed parameter tractable with genus as the parameter for 5-coloring n-vertex planar graphs in polylogarithmic rounds are provided. Expand
T
• Masař́ık, J. Muśılek, and O. Pangrác, Flexibility of triangle-free planar graphs, arXiv, 1902.02971
• 2019
List colourings of planar graphs
• M. Voigt
• Computer Science, Mathematics
• Discret. Math.
• 2006
A graph G=G(V,E) is called k-choosable if all lists L(v) have the cardinality k and G is L-list colourable for all possible assignments of such lists. Expand
Choosability in graphs
• Congr. Numer.
• 1980