• Corpus ID: 244773186

# Recolouring planar graphs of girth at least five

@inproceedings{Bartier2021RecolouringPG,
title={Recolouring planar graphs of girth at least five},
author={Valentin Bartier and Nicolas Bousquet and Carl Feghali and Marc Heinrich and Benjamin Moore and Th{\'e}o Pierron},
year={2021}
}
• Published 1 December 2021
• Mathematics
For a positive integer k , the k -recolouring graph of a graph G has as vertex set all proper k -colourings of G with two k -colourings being adjacent if they diﬀer by the colour of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the 7 -recolouring graphs of planar graphs, the 5 -recolouring graphs of triangle-free planar graphs and the 4 -recolouring graphs planar graphs of girth at least six are connected. On the other hand, there are planar…
6 Citations

## Figures and Tables from this paper

### 5-Coloring Reconfiguration of Planar Graphs with No Short Odd Cycles

• Mathematics
• 2022
The coloring reconﬁguration graph C k ( G ) has as its vertex set all the proper k -colorings of G , and two vertices in C k ( G ) are adjacent if their corresponding k -colorings diﬀer on a single

### Towards a conjecture on the recoloring diameter of planar graphs

• Mathematics
• 2022
A list assignment L of a graph G is a function that assigns to every vertex v of G a set L ( v ) of colors. A proper coloring α of G is called an L -coloring of G if α ( v ) ∈ L ( v ) for every v ∈ V

### Optimally Reconfiguring List and Correspondence Colourings

• Mathematics
ArXiv
• 2022
The reconﬁguration graph C k ( G ) for the k -colourings of a graph G has a vertex for each proper k -colouring of G , and two vertices of C k ( G ) are adjacent precisely when those k -colourings

### Short and local transformations between ($\Delta+1$)-colorings

• Mathematics
• 2022
Recoloring a graph is about finding a sequence of proper colorings of this graph from an initial coloring σ to a target coloring η. Each pair of consecutive colorings must differ on exactly one

### Short and local transformations between (Δ+1)-colorings

• Mathematics
ArXiv
• 2022
Recoloring a graph is about (cid:28)nding a sequence of proper colorings of this graph from an initial coloring σ to a target coloring η . Each pair of consecutive colorings must di(cid:27)er on

## References

SHOWING 1-10 OF 20 REFERENCES

### Mixing graph colourings

This thesis investigates some problems related to graph colouring, or, more precisely, graph re-colouring. Informally, the basic question addressed can be phrased as follows. Suppose one is given a

### Solving the Kotzig and Grünbaum problems on the separability of a cycle in planar graphs

• Mat. Zametki, 46:9–12
• 1989

### Islands in Graphs on Surfaces

• Mathematics
SIAM J. Discret. Math.
• 2016
Several bounds are proved on the size of islands in large graphs embeddable on fixed surfaces and every graph of genus g can be colored from lists of size 5, in such a way that each monochromatic component has size O(g).

### A three-color set for three-circle-free nets on the sphere

• science Z. Martin Luther Univ. Halle-Wittenberg, Math. Nat. Line, 8:109–120
• 1959

### Linear transformations between colorings in chordal graphs

• Mathematics
ESA
• 2019
It is proved that, as long as k ≥ d+4, there exists a transformation of length at most $f(\Delta) \cdot n$ between any pair of k-colorings of chordal graphs (where $\Delta$ denotes the maximum degree of the graph).

### An Update on Reconfiguring $10$-Colorings of Planar Graphs

• Mathematics
Electron. J. Comb.
• 2020
The number of colors is improved, showing that $R_{10}(G)$ has diameter at most $8n$ for every planar graph $G$ with $n$ vertices.