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作者机构:CNRS LaBRI Bordeaux France Univ Grenoble Alpes CNRS G SCOP Grenoble France Univ Bergen Dept Informat Bergen Norway Univ Durham Dept Comp Sci Durham England
出 版 物:《JOURNAL OF COMBINATORIAL THEORY SERIES B》 (组合理论杂志,B辑)
年 卷 期:2019年第135卷
页 面:179-199页
核心收录:
学科分类:07[理学] 0701[理学-数学] 070101[理学-基础数学]
基 金:Research Council of Norway
主 题:Graph colouring Kempe equivalence Regular graphs Kempe chain Wang-Swendsen-Kotecky algorithm Antiferromagnetic Potts model
摘 要:Let G be a graph with a vertex colouring alpha. 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 alpha 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 K-4 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-Kotecky algorithm for the antiferromagnetic Potts model at zero-temperature. (C) 2018 Elsevier Inc. All rights reserved.