In Most 6-regular Toroidal Graphs All 5-colorings are Kempe Equivalent

Abstract

A Kempe swap in a properly colored graph recolors one component of the subgraph induced by two colors, interchanging them on that component. Two k-colorings are Kempe k-equivalent if we can transform one into the other by a sequence of Kempe swaps, such that each intermediate coloring uses at most k colors. Meyniel proved that if G is planar, then all 5-colorings of G are Kempe 5-equivalent; this proof relies heavily on the fact that planar graphs are 5-degenerate. To prove an analogous result for toroidal graphs would require handling 6-regular graphs. That is the focus of this paper. We show that if G is a 6-regular graph that has an embedding in the torus with every non-contractible cycle of length at least 7, then all 5-colorings of G are Kempe 5-equivalent. Bonamy, Bousquet, Feghali, and Johnson asked specifically about the case that G is a triangulated toroidal grid, which is formed from the Cartesian product Cm Cn by adding a diagonal inside each 4-face, with all diagonals parallel. By slightly modifying the proof of our main result, we answer their question affirmatively when m ≥ 6 and n ≥ 6. IntroductionGiven a k-coloring ϕ of a graph G and colors α, β ∈ {1, . . . , k}, a Kempe swap Kempe swap recolors a component of the subgraph induced by colors α and β, interchanging those colors on that component, which is called a Kempe component Kempe component . Two k-colorings, ϕ 1 and ϕ 2 of a graph G are (Kempe) k-equivalent kequivalent

Ergodicity of the Wang–Swendsen–Kotecký algorithm on several classes of lattices on the torus

“…Result (a) is well known and can be found in [28]; results (b), (c), and (d) are mentioned without proof in Refs. [13], [19], and [6], respectively. By combining Corollary 3.2 and Theorem 2.3, we can deduce the following sufficient conditions for WSK-ergodicity: Corollary 3.3 (Ergodicity of WSK for graphs on the plane and the torus).…”

“…Also, (c) can be improved to q ≥ 2: if the graph is bipartite, then Theorem 2.6 proves ergodicity for q = 2; and if the graph is not bipartite, then there are no 2-colourings and the result holds vacuously. Finally, (d) is mentioned without proof in [6].…”

“…We tested several 6-regular triangulations G of the torus that are not 4-colourable (see the list in [6]), on the theory that these might be the most likely to yield counterexamples. All of the graphs we tried had a single 5-Kempe class, or there was no 5-colouring at all.…”

“…has a vertex in each face of G and an edge crossing each edge of G; it is also a connected graph on the torus. 6 We then obtain a bipartite quadrangulation Q = (V Q , E Q ) with vertex set V Q = V ∪ V * and edge set E Q that contain edges {i, j * } whenever i ∈ V lies on the boundary of the face of G that contains vertex j * ∈ V * . The graph Q = Q(G) is indeed a bipartite quadrangulation: on each face of Q, one pair of diametrically opposite vertices corresponds to an edge e ∈ E, and the other pair corresponds to the dual edge e * ∈ E * [see Fig.…”

“…On the other hand, WSK is not ergodic for q = 3 on periodic square lattices of size 3m × 3n with m, n relatively prime [29]. Similarly, for triangular lattices on the torus, the ergodicity of WSK has been proven for q ≥ 7 [22,34] (see Proposition 2.4), q = 6 [3], and q = 5 [6]; but, as mentioned above, ergodicity can fail for q = 4 [35]. Finally, for kagome lattices on the torus, the WSK-ergodicity has been proven for q ≥ 5 [22,34] (see Proposition 2.4) and q = 4 [3,32]; but it can fail for q = 3 [36].…”