Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces

@article{Postle2019LinearTimeAE,
  title={Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces},
  author={Luke Postle},
  journal={2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)},
  year={2019},
  pages={929-941}
}
  • L. Postle
  • Published 7 April 2019
  • Computer Science, Mathematics
  • 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such colorings. Here, we provide the first linear-time algorithms to find such colorings. For a fixed surface S, Thomassen showed in 1997 that there exists a linear-time algorithm to decide if a graph embedded in S is 5-colorable and similarly in 2003 if a graph of girth… Expand
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References

SHOWING 1-10 OF 34 REFERENCES
3-List-coloring graphs of girth at least five on surfaces
  • L. Postle
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 2021
TLDR
A linear isoperimetric bound for 3-list-coloring graphs of girth at least five is proved, and many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. Expand
Optimal Distributed Coloring Algorithms for Planar Graphs in the LOCAL model
TLDR
A novel algorithm based on a novel technique that detects small structures that can be easily colored given a proper coloring of the rest of the vertices and removes them from the graph until the graph contains a small enough number of edges is presented. Expand
Three-coloring triangle-free planar graphs in linear time
TLDR
A linear-time algorithm to find a 3-coloring of a given triangle-free planar graph that avoids using any complex data structures, which makes it easy to implement and gives a yet simpler proof of Grötzsch's theorem. Expand
Distributed Coloring in Sparse Graphs with Fewer Colors
TLDR
Among other results, it is shown that no distributed algorithm can color every n-vertex planar graph with 4 colors in o(n) rounds, and bounds on the number of colors turn out to be quite sharp in general. Expand
List-coloring embedded graphs
For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in timeExpand
5-LIST-COLORING GRAPHS ON SURFACES
setting. In Section 5.8, we apply the general theory to the family of 6-listcritical graphs to derive the main results for 5-list-coloring. Finally, in Section 5.9, we apply the theory for a slightlyExpand
A not 3-choosable planar graph without 3-cycles
  • M. Voigt
  • Computer Science, Mathematics
  • Discret. Math.
  • 1995
TLDR
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
Five-list-coloring graphs on surfaces II. A linear bound for critical graphs in a disk
TLDR
This work proves a conjecture of Dvořak et al. that if H is a minimal sub graph of G such that C is a subgraph of H and every L-coloring of C that extends to an L- Coloring of H also extends to a L-Coloring of G, then V ( H) is a list coloring graphs on surfaces that satisfy an isoperimetric inequality suggested by this result. Expand
Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition
TLDR
The first sublogarithmic algorithm for computing an MIS on graphs of bounded arboricity is devised, which demonstrates that this methodology is very powerful and shows nearly-tight lower bounds on the running time of any distributed algorithms for computing a forests-decomposition. Expand
List colourings of planar graphs
  • M. Voigt
  • Computer Science
  • Discret. Math.
  • 1993
TLDR
A graph G is k-choosable if all lists L(u) have the cardinality k and G is L-list colourable for all possible assignments of such lists. Expand
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