# A unified approach to distance-two colouring of graphs on surfaces

@article{Amini2013AUA, title={A unified approach to distance-two colouring of graphs on surfaces}, author={Omid Amini and Louis Esperet and Jan van den Heuvel}, journal={Combinatorica}, year={2013}, volume={33}, pages={253-296} }

AbstractIn this paper we introduce the notion of Σ-colouring of a graph G: For given subsets Σ(v) of neighbours of v, for every v∈V (G), this is a proper colouring of the vertices of G such that, in addition, vertices that appear together in some Σ(v) receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic…

## 20 Citations

Degeneracy and Colorings of Squares of Planar Graphs without 4-Cycles

- MathematicsComb.
- 2020

It is shown that 4-cycles are unique in having this property, and the upper bounds on each of these parameters of G 2 can all be lowered to Δ( G ) + 2 (which is best possible).

Edge-colouring graphs with local list sizes

- Mathematics
- 2020

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the…

Induction Schemes : From Language Separation to Graph Colorings. (Schémas d'induction : from languages separation to graph colorings)

- Mathematics, Computer Science
- 2019

Two types of contributions are presented: a generalization of a decidability result to the setting of infinite words, together with lower bounds for the complexity of the separation problem.

Coloring Jordan Regions and Curves

- MathematicsSIAM J. Discret. Math.
- 2017

It is shown that the elements of $\mathcal{F}$ can be colored with at most k+1 colors so that intersecting Jordan regions are assigned distinct colors, which can be used to bound the ratio between the maximum number of vertex-disjoint directed cycles in a planar digraph, and its fractional counterpart.

Third Case of the Cyclic Coloring Conjecture

- MathematicsSIAM J. Discret. Math.
- 2015

The case Δ⁎=6 of the Cyclic Coloring Conjecture is proved, which says that every graph that can be drawn in the plane with each edge crossed by at most one other edge is 6-colorable.

Coloring pseudo-disks and Jordan curves

- MathematicsArXiv
- 2016

It is shown that the elements of $\mathcal{F}$ can be colored with at most $k+1$ colors so that intersecting pseudo-disks are assigned distinct colors, which can be used to bound the ratio between the maximum number of vertex-disjoint directed cycles in a planar digraph, and its fractional counterpart.

Nonrepetitive graph colouring

- MathematicsThe Electronic Journal of Combinatorics
- 2021

The goal is to give a unified and comprehensive presentation of the major results and proof methods, as well as to highlight numerous open problems about nonrepetitive colourings of graphs.

Square coloring planar graphs with automatic discharging

- Computer Science, MathematicsArXiv
- 2022

This paper uses a Linear Programming approach to automatically look for a discharging proof, and makes some progress towards Wegner’s conjecture for distance- 2 coloring of planar graphs, by showing that 12 colors arecient to color at distance 2 every planar graph with maximum degree 4.

## References

SHOWING 1-10 OF 44 REFERENCES

Asymptotics of the list-chromatic index for multigraphs

- MathematicsRandom Struct. Algorithms
- 2000

The list-chromatic index, χ′ l (G) of a multigraph G, is the least t such that if S(A) is a set of size t for each A ∈ E := E(G), then there exists a proper coloring σ of G with σ(A), which is at least the fractional chromatic index.

Coloring powers of planar graphs

- MathematicsSODA '00
- 2000

In general, it is shown for a fixed integer $k\geq1$ the inductiveness, the chromatic number, and the choosability of Gk to be $O(\Delta^{\lfloor k/2 \rfloor})$, which is tight.

A new upper bound on the cyclic chromatic number

- MathematicsJ. Graph Theory
- 2007

This research was done during visits of OVB, AG, and JvdH to the University of Twente, while HJB was employed there, and the bound was improved to 5 ∗ by Borodin, Sanders, and Zhao.

A bound on the chromatic number of the square of a planar graph

- MathematicsJ. Comb. Theory, Ser. B
- 2005

Graphs on Surfaces

- MathematicsJohns Hopkins series in the mathematical sciences
- 2001

This chapter discusses Embeddings Combinatorially, Contractibility, of Cycles, and the Genus Problem, which focuses on planar graphs and the Jordan Curve Theorem, and colorings of Graphs on Surfaces, which are 5-choosable.

Maximum matching and a polyhedron with 0,1-vertices

- Mathematics
- 1965

The emphasis in this paper is on relating the matching problem to the theory of continuous linear programming, and the algorithm described does not involve any "blind-alley programming" -which, essentially, amounts to testing a great many combinations.

Graph Minors. XVI. Excluding a non-planar graph

- MathematicsJ. Comb. Theory, Ser. B
- 2003