# Building a larger class of graphs for efficient reconfiguration of vertex colouring

@article{Biedl2020BuildingAL, title={Building a larger class of graphs for efficient reconfiguration of vertex colouring}, author={Therese C. Biedl and Anna Lubiw and Owen D. Merkel}, journal={ArXiv}, year={2020}, volume={abs/2003.01818} }

A $k$-colouring of a graph $G$ is an assignment of at most $k$ colours to the vertices of $G$ so that adjacent vertices are assigned different colours. The reconfiguration graph of the $k$-colourings, $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. For a $k$-colourable graph $G$, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$. It is…

## 2 Citations

C O ] 2 A ug 2 02 1 Mixing colourings in 2 K 2-free graphs

- 2021

The reconfiguration graph for the k-colourings of a graph G, denoted Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge if they differ in colour on…

Recolouring weakly chordal graphs and the complement of triangle-free graphs

- Mathematics, Computer ScienceDiscrete Mathematics
- 2022

It is proved that for all n ≥ 1, there exists a k-colourable weakly chordal graph G where Rk+n(G) is disconnected, answering an open question of Feghali and Fiala.

## References

SHOWING 1-10 OF 66 REFERENCES

Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs

- Mathematics, Computer ScienceDiscret. Math.
- 2020

This paper answers a question of Bonamy, Johnson, Lignos, Patel and Paulusma by constructing for each k-colourable weakly chordal graph G such that $R_{k+1}(G)$ is disconnected.

Paths between colourings of sparse graphs

- Mathematics, Computer ScienceEur. J. Comb.
- 2019

The proof can be transformed into a simple polynomial time algorithm that finds a path between a given pair of colourings in the reconfiguration graph R_k(G).

Reconfiguring 10-Colourings of Planar Graphs

- Mathematics, Computer ScienceGraphs Comb.
- 2020

It is given a simple proof that if G is a planar graph on n vertices, then R 10 ( G ) has diameter at most n(n + 1)/ 2 and this affirms Cereceda’s conjecture for planar graphs in the case of 2k = 2k ℓ = 2 k .

Towards Cereceda's conjecture for planar graphs

- Mathematics, Computer ScienceJ. Graph Theory
- 2020

The proof of Cereceda's conjecture that, for every $k$-degenerate graph G on $n$ vertices, $R_{k+2}(G)$ has diameter $\mathcal{O}({n^2})$ is improved to improve this bound for planar graphs to $2^{\mathcal(O)({sqrt{n}})}$.

Reconfiguring colourings of graphs with bounded maximum average degree

- Computer Science, MathematicsJ. Comb. Theory, Ser. B
- 2021

It is proved that for every graph G with n vertices and maximum average degree d - \epsilon, the reconfiguration graph R_k(G) has diameter O(n(\log n)^{d})$.

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

- Computer Science, MathematicsElectron. 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.

Kempe equivalence of colourings of cubic graphs

- Computer Science, MathematicsEur. J. Comb.
- 2017

This paper addresses the case of k=3 by showing that all 3-colourings of a cubic graph G are Kempe equivalent unless $G$ is the complete graph $K_4$ or the triangular prism.

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. B
- 2019

It is proved that the conjecture that all-colourings of a cubic graph that is neither $K_4$ nor the triangular prism are Kempe equivalent holds for each $k\geq 4$.

Recoloring graphs via tree decompositions

- Mathematics, Computer ScienceEur. J. Comb.
- 2018

It is proved that the shortest sequence between any two $(tw+2)$-colorings is at most quadratic (which is optimal up to a constant factor), a problem left open in Bonamy et al. (2012).

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

- Mathematics, Computer ScienceJ. Graph Theory
- 2016

This work shows an analogue of Brooks' Theorem by proving that from any $k-colouring, $k>\Delta$, a $\Delta$-colours of G can be obtained by a sequence of $O(n^2) recolourings using only the original $k$ colours.