# Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs

@inproceedings{Dabrowski2015FillingTC,
title={Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs},
author={Konrad Dabrowski and François Dross and Matthew Johnson and Dani{\"e}l Paulusma},
booktitle={IWOCA},
year={2015}
}
• Published in IWOCA 22 June 2015
• Mathematics
A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots \}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a subset of $\{1, 2, \dots\}$ of size $k$. A colouring $c$ of $G$ respects a $k$-regular list assignment $L$ of $G$ if $c(u)\in L(u)$ for every $u\in V$. A graph $G$ is $k$-choosable if for every $k$-regular list assignment $L$ of $G$, there exists a colouring…
9 Citations

## Figures from this paper

A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs
• Mathematics
MFCS
• 2019
A dichotomy result is obtained stating that for fixed $k, the b-Coloring problem is polynomial-time solvable whenever$p \in \{0, 1\}$and, even when$k = 3, it is NP-complete whenever $p \ge 2$.
Filling the complexity gaps for colouring planar and bounded degree graphs
• Mathematics
J. Graph Theory
• 2019
Using known examples of non‐3‐choosable and non‐4‐choOSable graphs, this enables the complexity ofk‐REGULARLISTCOLOURING restricted to planar graphs, planar bipartite graphs,planar triangle‐free graphs, and planar graph with no4‐cycles and no5‐cycles to be classified.
Open Problems on Graph Coloring for Special Graph Classes
This work surveys known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded and considers coloring variants, such as precoloring extensions and list colorings and gives some open problems in the area of on-line coloring.
Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter
• Mathematics
ArXiv
• 2021
This work examines to what extent the situation may change if in addition the input graph has bounded diameter, and concludes that the problem is NP-complete even for graphs of arbitrarily large fixed girth.
List-coloring - Parameterizing from triviality
• Mathematics
Theor. Comput. Sci.
• 2020
Some (in)tractable Parameterizations of Coloring and List-Coloring
• Mathematics
FAW
• 2018
This work considers a few versions of the problems that are polynomial time solvable, and tries to extend the notion of feasible algorithms by parameterizing suitably in the paradigm of parameterized complexity.
Fully Dynamic (Δ +1)-Coloring in O(1) Update Time
• Mathematics, Computer Science
ACM Transactions on Algorithms
• 2022
An improved randomized algorithm for (Δ +1)-coloring that achieves O(1) amortized update time and it is shown that this bound holds not only in expectation but also with high probability.
Vertex partition of sparse graphs
Le Theoreme des Quatre Couleurs, conjecture en 1852 et prouve en 1976, est a l'origine de l'etude des partitions des sommets de graphes peu denses. Il affirme que toute carte plane peut etre coloriee

## References

SHOWING 1-10 OF 61 REFERENCES
Closing Complexity Gaps for Coloring Problems on H-Free Graphs
• Mathematics
ISAAC
• 2012
The Precoloring Extension problem and the l-List Coloring problem for H-free graphs are classified and it is shown that List 4-Coloring is NP-complete for P 6- free graphs, where P 6 is the path on six vertices.
A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs
• Mathematics
J. Graph Theory
• 2017
This work surveys known results on the computational complexity of Colouring and \$k-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs, and considers a number of variants.
On 3-colorable non-4-choosable planar graphs
• Mathematics
• 1997
An L-list coloring of a graph G is a proper vertex coloring in which every vertex v gets a color from a list L(v) of allowed colors. G is called k-choosable if all lists L(v) have exactly k elements
Algorithmic complexity of list colorings
• Mathematics
Discret. Appl. Math.
• 1994
List colourings of planar graphs