On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Bonamy, Marthe; Bousquet, Nicolás; Feghali, Carl; Johnson, Matthew

doi:10.1016/j.jctb.2018.08.002

“…Conjecture 1 was disproven for the case k = 3 by Feghali, Johnson and Paulusma [34], who showed that all 3-colourings of a connected 3-regular graph G are Kempe equivalent unless G is isomorphic to K 4 or the triangular prism. For k ≥ 4, Conjecture 1 was proven by Bonamy, Bousquet, Feghali, and Johnson [6].…”

Section: Conjecture 1 ([50]mentioning

confidence: 95%

Building a Larger Class of Graphs for Efficient Reconfiguration of Vertex Colouring

Biedl

¹

,

Lubiw

²

,

Merkel

³

2021

Trends in Mathematics

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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, 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. For a k-colourable graph G, we investigate the connectivity and diameter of R k+1 (G). It is known that not all weakly chordal graphs have the property that R k+1 (G) is connected. 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.

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“…The conjecture was proved for k ≥ 4 [98], and eventually it was shown that the triangular prism and K 3 are the only 3-regular graphs for which the reconfiguration graph for k = 3 is not connected [99].…”

Section: Variants Of Coloringmentioning

confidence: 99%

Introduction to Reconfiguration

Nishimura

¹

2018

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

show abstract

“…The conjecture was proved for k ≥ 4 [98], and eventually it was shown that the triangular prism and K 3 are the only 3-regular graphs for which the reconfiguration graph for k = 3 is not connected [99].…”

Section: Variants Of Coloringmentioning

confidence: 99%

Introduction to Reconfiguration

Nishimura¹

2017

Preprint

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

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On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Cited by 34 publications

References 34 publications

Building a Larger Class of Graphs for Efficient Reconfiguration of Vertex Colouring

Building a Larger Class of Graphs for Efficient Reconfiguration of Vertex Colouring

Introduction to Reconfiguration

Introduction to Reconfiguration

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