Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs

belcastro, sarah-marie; Haas, Ruth

doi:10.1016/j.disc.2014.02.014

Cited by 16 publications

(34 citation statements)

References 8 publications

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“…Mohar [46] showed that if k ≥ χ ′ (G) + 2, then EC k (G) is connected, i.e., In [5], belcastro and Haas provided partial answers to Mohar's question on cubic bipartite graphs G with K ′ (G, 3) = 1. They showed that all 3-edge-colourings of planar bipartite cubic graphs are edge-Kempe equivalent, and constructed infinite families of simple nonplanar 3connected bipartite cubic graphs, all of whose 3-edge-colourings are edge-Kempe equivalent.…”

Section: The K-edge-colouring Graphmentioning

confidence: 99%

Reconfiguration of Colourings and Dominating Sets in Graphs

Nasserasr¹

2019

50 Years of Combinatorics, Graph Theory, and Computing

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We survey results concerning reconfigurations of colourings and dominating sets in graphs. 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. Similarly, the vertices of the k-edge-colouring graph EC k (G) of g are the proper k-edge-colourings of G, where two k-edge-colourings are adjacent if one can be obtained from the other by switching two colours along an edge-Kempe chain, i.e., a maximal two-coloured alternating path or cycle of edges.The vertices of the k-dominating graph D k (G) are the (not necessarily minimal) dominating sets of G of cardinality k or less, two dominating sets being adjacent in D k (G) if one can be obtained from the other by adding or deleting 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 along edges or not.We consider these and related types of colouring and domination reconfiguration graphs. Conjectures, questions and open problems are stated within the relevant sections. * Supported by the Natural Sciences and Engineering Research Council of Canada.1 We use the term connectedness instead of connectivity when referring to the question of whether a graph is connected or not, as the latter term refers to a specific graph parameter.

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Section: The K-edge-colouring Graphmentioning

confidence: 99%

Reconfiguration of Colourings and Dominating Sets in Graphs

Nasserasr¹

2019

50 Years of Combinatorics, Graph Theory, and Computing

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“…It is known [2] that G admits two edge-Kempe inequivalent colorings c 1 and c 2 . The graph G is 3-regular.…”

Section: An Examplementioning

confidence: 99%

“…Two proper edge colorings c 1 and c 2 are edge Kempe equivalent if G c ( , ) 2 can be obtained from G c ( , ) 1 by a sequence of edge Kempe switches. We denote this by G c G c ( , )~( , ) 1 2 .…”

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confidence: 99%

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Edge Kempe equivalence of regular graph covers

Lazarovich

Levit

2019

Journal of Graph Theory

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Let G be a finite d‐regular graph with a proper edge coloring. An edge Kempe switch is a new proper edge coloring of G obtained by switching the two colors along some bichromatic cycle. We prove that any other edge coloring can be obtained by performing finitely many edge Kempe switches, provided that G is replaced with a suitable finite covering graph. The required covering degree is bounded above by a constant depending only on d.

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“…When periodic boundary conditions are used, this dynamics is known to be nonergodic, for reasons not fully understood [2][3][4][5]9]. The color configurations can be put in equivalence classes, called Kempe sectors, if they are connected by the dynamics, the number of which, n K > 1 [4,5], has been enumerated numerically on small random cubic graphs [16], or regular hexagonal clusters [4,17]. An invariant has been found, allowing to distinguish odd from even colorings [4,5,17].…”

Section: Introductionmentioning

confidence: 99%

Invariants of winding-numbers and steric obstruction in dynamics of flux lines

Cepas

Akhmetiev

2019

SciPost Phys.

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We classify the sectors of configurations that result from the dynamics of 2d crossing flux lines, which are the simplest degrees of freedom of the 3-coloring lattice model. We show that the dynamical obstruction is the consequence of two effects: (i) conservation laws described by a set of invariants that are polynomials of the winding numbers of the loop configuration, (ii) steric obstruction that prevents paths between configurations, for lack of free space. We argue that the invariants fully classify the configurations in five, chiral and achiral, sectors and no further obstruction in the limit of low-winding numbers.

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Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs

Cited by 16 publications

References 8 publications

Reconfiguration of Colourings and Dominating Sets in Graphs

Reconfiguration of Colourings and Dominating Sets in Graphs

Edge Kempe equivalence of regular graph covers

Invariants of winding-numbers and steric obstruction in dynamics of flux lines

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