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On the diameter of reconfiguration graphs for vertex colourings

2011
*
Electronic Notes in Discrete Mathematics
*

Lower Bounds We can show that

doi:10.1016/j.endm.2011.09.028
fatcat:agrqqtrpbjasjphw62tpvpextq
*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. ...##
###
Reconfiguring 10-colourings of planar graphs
[article]

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. ...

##
###
Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

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. ...

##
###
Paths between colourings of sparse graphs
[article]

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. ...

##
###
A polynomial version of Cereceda's conjecture
[article]

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. ...

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

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*...

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

2015
*
arXiv
*
pre-print

We use this result to study

arXiv:1501.05800v1
fatcat:ncfbafdi6behhoypg2nkn7rqzu
*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. ...##
###
A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

2015
*
Journal of Graph Theory
*

Let ∆ ≥ 3 and assume that we have an O(n 2 )-time algorithm

doi:10.1002/jgt.22000
fatcat:t6j3rwuddvhvjhe6544hnvybbu
*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 ). ...##
###
A Reconfigurations Analogue of Brooks' Theorem
[chapter]

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 ). ...

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

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*. ...

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

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. ...

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

2022
*
Electronic Journal of Combinatorics
*

As a side result, we also obtain a subexponential bound

doi:10.37236/9455
fatcat:ztsftgjfs5a37crsxc64dninzq
*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. ...##
###
Reconfiguration of Dominating Sets
[chapter]

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. ...

##
###
Reconfiguration of dominating sets

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. ...

##
###
Reconfiguration of Dominating Sets
[article]

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. ...

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