Kempe Equivalence Research Articles

Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective Plane

Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective Plane

Conical $SL(3)$ foams

In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighborhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighborhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straightforward to show that modules associated to the dodecahedron graph are free of rank 60, which is still an open problem for the original unoriented SL(3) foam theory.

In most 6-regular toroidal graphs all 5-colorings are Kempe equivalent

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two k-colorings are k-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, T[m×n], is formed from (a toroidal embedding of) the Cartesian product of Cm and Cn by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of T[m×n] are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of T[m×n] are 6-equivalent. They asked whether the same is true for 5-colorings. We answer their question affirmatively when m,n≥6. Further, we show that if G is 6-regular with a toroidal embedding where every non-contractible cycle has length at least 7, then all 5-colorings of G are 5-equivalent. Our results relate to the antiferromagnetic Pott’s model in statistical mechanics.

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Let G be a graph with a vertex colouring α. Let a and b be two colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. A colouring of G obtained from α by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes.A conjecture of Mohar (2007) asserts that, for k≥3, all k-colourings of a k-regular graph that is not complete are Kempe equivalent. It was later shown that all 3-colourings of a cubic graph that is neither K4 nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each k≥4. We also report the implications of this result on the validity of the Wang–Swendsen–Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.

Kempe Equivalence of Colorings of 4-regular Graphs

Given a graphG and a proper vertex coloring ofG, a 2-coloring induced subgraph ofG is a subgraph induced by all the vertices with one of two colors, a component of a 2-coloring induced subgraph is called a 2-coloring component. To make a Kempe change is to obtain one coloring from another by exchanging the colors of vertices in a 2-coloring component. Two colorings are Kempe equivalent if each one can be obtained from the other by a series of Kempe changes. Mohar conjectured that, for k3, all k-colorings of connected k-regular graphs that are not complete are Kempe equivalent. Feghali et al. addressed the case k=3, and it is still an unsolved conjecture for k4. This paper considers the casek=4 by showing that: (1) ifG is a connected 4-regular graph that is not 3-connected, then all 4-colorings ofG are Kempe equivalent; (2) ifG is a connected 4-regular graph that contains an induced subgraph isomorphic to a 4-wheel or a nearly complete graph of order 5, then all 4-colorings ofG are Kempe equivalent; (3) ifG is a 3-connceted 4-regular graph with a 4-coloringf and a vertexx such that there are three or four neighbors ofx colored with the same color under f, then all 4-colorings ofG are Kempe equivalent.

Kempe equivalence of colourings of cubic graphs

Given a graph G=(V,E) and a proper vertex colouring of G, a Kempe chain is a subset of V that induces a maximal connected subgraph of G in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for k≥3, all k-colourings of connected k-regular graphs that are not complete are Kempe equivalent. We address the case k=3 by showing that all 3-colourings of a connected cubic graph G are Kempe equivalent unless G is the complete graph K4 or the triangular prism.

Kempe Equivalence of Colourings of Cubic Graphs

Given a graph G=(V,E) and a proper vertex colouring of G, a Kempe chain is a subset of V that induces a maximal connected subgraph of G in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for k≥3, all k-colourings of connected k-regular graphs that are not complete are Kempe equivalent. We address the case k=3 by showing that all 3-colourings of a connected cubic graph G are Kempe equivalent unless G is the complete graph K4 or the triangular prism.

Solution of Vizing's Problem on Interchanges for the case of Graphs with Maximum Degree 4 and Related Results

Let G be a Class 1 graph with maximum degree 4 and let be an integer. We show that any proper t-edge coloring of G can be transformed to any proper 4-edge coloring of G using only transformations on 2-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper 5-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent.

Kempe Equivalence of Edge‐Colorings in Subcubic and Subquartic Graphs

AbstractIt is proved that all 4‐edge‐colorings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree Δ = 3 is a threshold for Kempe equivalence of (Δ+1)‐edge‐colorings, as such an equivalence does not hold in general when Δ = 4. One extra color allows a similar result in this latter case; however, namely, when Δ≤4 it is shown that all (Δ+2)‐edge‐colorings are Kempe equivalent. © 2011 Wiley Periodicals, Inc. J Graph Theory

A new Kempe invariant and the (non)-ergodicity of the Wang–Swendsen–Kotecký algorithm

We prove that for the class of three-colorable triangulations of a closed-oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L, 3M) of the torus with 3 ⩽ L ⩽ M, there are at least two Kempe equivalence classes. This result implies, in particular, that the Wang–Swendsen–Kotecký algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L, 3M) of the torus is not ergodic.

Some Topological Methods in Graph Coloring Theory

Abstract Attempts to solve the famous Four Color Problem led to fruitful discoveries and rich coloring theories. In this talk, some old and some recent applications of tools from topology to graph coloring problems will be presented. In particular, the following subjects will be treated: The use of Euler's formula and local planarity conditions, Kempe equivalence, homotopy, winding number and its higher dimensional analogues.

The NP-Completeness of Edge-Coloring

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.