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On the diameter of reconfiguration graphs for vertex colourings

Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, Daniël Paulusma
2011 Electronic Notes in Discrete Mathematics  
Lower Bounds We can show that the 3-colour diameter of a path on n vertices is Θ(n 2 ) and have an example of a k-colourable chordal graph with (k + 1)-colour diameter of order Θ(n 2 ) for every k ≥ 4.  ...  Theorem 3.4 The class of chordal bipartite graphs is 2-colour-dense.  ...  We define the -colour diameter of a graph G to be the diameter of the reconfiguration graph of -colourings of G. Theorem 2.1 For an integer k ≥ 1, let G be a k-colour-dense graph class.  ... 
doi:10.1016/j.endm.2011.09.028 fatcat:agrqqtrpbjasjphw62tpvpextq

Reconfiguring 10-colourings of planar graphs [article]

Carl Feghali
2019 arXiv   pre-print
The reconfiguration graph R_k(G) of the k-colourings of a graph G has as vertex set the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex.  ...  A conjecture of Cereceda from 2007 asserts that for every integer ℓ≥ k + 2 and k-degenerate graph G on n vertices, R_ℓ(G) has diameter O(n^2). The conjecture has been verified only when ℓ≥ 2k + 1.  ...  Acknowledgements The author is grateful to Louis Esperet for pointing out Corollary 2. This work is supported by the Research Council of Norway via the project CLAS-SIS.  ... 
arXiv:1902.02278v2 fatcat:lzqj7jhqxvflha3sbnx4mhj7uq

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, Daniël Paulusma
2012 Journal of combinatorial optimization  
The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G.  ...  We show that for each k-colour-dense graph G, the reconfiguration graph of the -colourings of G is connected and has diameter O(|V | 2 ), for all ≥ k + 1.  ...  We define the -colour diameter of a graph G to be the diameter of R G . Theorem 2. For an integer k ≥ 1, let G be a k-colour-dense graph on n vertices.  ... 
doi:10.1007/s10878-012-9490-y fatcat:n3u3suhenffovaenkhoywp2r6u

Paths between colourings of sparse graphs [article]

Carl Feghali
2020 arXiv   pre-print
The reconfiguration graph R_k(G) of the k-colourings of a graph G has as vertex set the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex.  ...  For every ϵ > 0 and every graph G with n vertices and maximum average degree d - ϵ, there exists a constant c = c(d, ϵ) such that R_k(G) has diameter O(n^c).  ...  Acknowledgements The author is grateful to the referees for carefully checking the paper and to a referee whose comments have helped to improve the exposition of the paper.  ... 
arXiv:1803.03950v2 fatcat:3dqnkgvrbzatdm2r3fsf2v45zq

A polynomial version of Cereceda's conjecture [article]

Nicolas Bousquet, Marc Heinrich
2019 arXiv   pre-print
The $k$-reconfiguration graph of $G$ is the graph whose vertices are the proper $k$-colourings of $G$, with an edge between two colourings if they differ on exactly one vertex.  ...  In this paper, we prove that the diameter of the $k$-reconfiguration graph of a $d$-degenerate graph is $O(n^{d+1})$ for $k \ge d+2$.  ...  Clearly H is still planar, and it is still bipartite since v and w are on the same side of the bipartition of G. Moreover, α and β are proper colourings of H.  ... 
arXiv:1903.05619v1 fatcat:fyxmdpmazbbovddibraomkgqhi

Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs [article]

Carl Feghali, Jiří Fiala
2019 arXiv   pre-print
The reconfiguration graph R_k(G) of the k-colourings of a graph G contains as its vertex set the k-colourings of G and two colourings are joined by an edge if they differ in colour on just one vertex of  ...  We also introduce a subclass of k-colourable weakly chordal graphs which we call k-colourable compact graphs and show that for each k-colourable compact graph G on n vertices, R_k+1(G) has diameter O(n  ...  The reconfiguration graph R k (G) of the k-colourings of G has as vertex set the set of all k-colourings of G and two vertices of R k (G) are adjacent if they differ on the colour of exactly one vertex  ... 
arXiv:1902.08071v2 fatcat:fg57lvbsarduhcpx3rccjmlczm

A Reconfigurations Analogue of Brooks' Theorem and its Consequences [article]

Carl Feghali, Matthew Johnson, Daniël Paulusma
2015 arXiv   pre-print
We use this result to study the reconfiguration graph R_k(G) of the k-colourings of G.  ...  The vertex set of R_k(G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex.  ...  Let ∆ ≥ 3 and assume that we have an O(n 2 )-time algorithm for connected (∆ − 2)-degenerate graphs on n vertices with maximum degree ∆ − 1.  ... 
arXiv:1501.05800v1 fatcat:ncfbafdi6behhoypg2nkn7rqzu

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

Carl Feghali, Matthew Johnson, Daniël Paulusma
2015 Journal of Graph Theory  
Let ∆ ≥ 3 and assume that we have an O(n 2 )-time algorithm for connected (∆ − 2)-degenerate graphs on n vertices with maximum degree ∆ − 1.  ...  We use induction on ∆. If ∆ ∈ {1, 2} the statement is trivially true.  ...  For any pair of integers d, k with k ≥ d + 2, the reconfiguration graph R k (G) of a d-degenerate graph G has diameter O(n 2 ).  ... 
doi:10.1002/jgt.22000 fatcat:t6j3rwuddvhvjhe6544hnvybbu

A Reconfigurations Analogue of Brooks' Theorem [chapter]

Carl Feghali, Matthew Johnson, Daniël Paulusma
2014 Lecture Notes in Computer Science  
The vertex set of R k (G) is the set of all possible kcolourings of G and two colourings are adjacent if they differ on exactly one vertex.  ...  To recolour G is to obtain a new proper colouring by changing the colour of one vertex.  ...  Cereceda [9] conjectured that the diameter of the reconfiguration graph on (k +2)-colourings of a k-degenerate graph on n vertices is O(n 2 ).  ... 
doi:10.1007/978-3-662-44465-8_25 fatcat:owverzuc3fhyzoaxzt2vbraqeu

Reconfiguration of Colourings and Dominating Sets in Graphs: a Survey [article]

C.M. Mynhardt, S. Nasserasr
2020 arXiv   pre-print
The vertices of the k-colouring graph C_k(G) of a graph G correspond to the proper k-colourings of a graph G, with two k-colourings being adjacent whenever they differ in the colour of exactly one vertex  ...  On the other hand, when we restrict the dominating sets to be minimum dominating sets, for example, we obtain different types of domination reconfiguration graphs, depending on whether vertices are exchanged  ...  in the colour of exactly one vertex.  ... 
arXiv:2003.05956v1 fatcat:b2akpiblpvg6zk3n55ydlcp64y

Building a larger class of graphs for efficient reconfiguration of vertex colouring [article]

Therese Biedl, Anna Lubiw, Owen Merkel
2020 arXiv   pre-print
The reconfiguration graph of the k-colourings, R_k(G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge in R_k(G) if they differ in colour on exactly one vertex  ...  On the other hand, R_k+1(G) is connected and of diameter O(n^2) for several subclasses of weakly chordal graphs such as chordal, chordal bipartite, and P_4-free graphs.  ...  In Section 2.1 we discuss known results on reconfiguration of vertex colouring for these graph classes.  ... 
arXiv:2003.01818v1 fatcat:pbfrhhfohjdx3pqjiuwvleivlm

Decreasing the Maximum Average Degree by Deleting an Independent Set or a $d$-Degenerate Subgraph

Wojciech Nadara, Marcin Smulewicz
2022 Electronic Journal of Combinatorics  
As a side result, we also obtain a subexponential bound on the diameter of reconfiguration graphs of generalized colourings of graphs with bounded value of their $\mathrm{mad}$.  ...  The maximum average degree $\mathrm{mad}(G)$ of a graph $G$ is the maximum over all subgraphs of $G$, of the average degree of the subgraph.  ...  We would also like to thank Bartosz Walczak for discussions on this topic and anonymous referees of the Electronic Journal of Combinatorics for their valuable comments.  ... 
doi:10.37236/9455 fatcat:ztsftgjfs5a37crsxc64dninzq

Reconfiguration of Dominating Sets [chapter]

Akira Suzuki, Amer E. Mouawad, Naomi Nishimura
2014 Lecture Notes in Computer Science  
On the positive side, we show that Dn−µ(G) is connected and of linear diameter for any graph G on n vertices with a matching of size at least µ + 1.  ...  For various values of k, we consider properties of D k (G), the graph consisting of a vertex for each dominating set of size at most k and edges specified by the adjacency relation.  ...  Determining the diameter of the reconfiguration graph will result in an upper bound on the length of any reconfiguration sequence.  ... 
doi:10.1007/978-3-319-08783-2_35 fatcat:xt5bijpazfc4xhxchombuta7me

Reconfiguration of dominating sets

Akira Suzuki, Amer E. Mouawad, Naomi Nishimura
2015 Journal of combinatorial optimization  
On the positive side, we show that Dn−µ(G) is connected and of linear diameter for any graph G on n vertices with a matching of size at least µ + 1.  ...  For various values of k, we consider properties of D k (G), the graph consisting of a vertex for each dominating set of size at most k and edges specified by the adjacency relation.  ...  Determining the diameter of the reconfiguration graph will result in an upper bound on the length of any reconfiguration sequence.  ... 
doi:10.1007/s10878-015-9947-x fatcat:iq6rjeongnhephjksvdusv3np4

Reconfiguration of Dominating Sets [article]

Akira Suzuki, Amer E. Mouawad, Naomi Nishimura
2014 arXiv   pre-print
On the positive side, we show that D_n-m(G) is connected and of linear diameter for any graph G on n vertices having at least m+1 independent edges.  ...  For various values of k, we consider properties of D_k(G), the graph consisting of a vertex for each dominating set of size at most k and edges specified by the adjacency relation.  ...  Determining the diameter of the reconfiguration graph will result in an upper bound on the length of any reconfiguration sequence.  ... 
arXiv:1401.5714v2 fatcat:43cvcehofnak5nkp3td2cjwurq
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