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Finding Paths between Graph Colourings: Computational Complexity and Possible Distances

Bonsma, Paul; Cereceda, Luis; van den Heuvel, Jan; Johnson, Matthew


Paul Bonsma

Luis Cereceda

Jan van den Heuvel


Suppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? We prove a dichotomy theorem for the computational complexity of this decision problem: for values of k⩽3 the problem is polynomial-time solvable, while for any fixed k⩾4 it is PSPACE-complete. What is more, we establish a connection between the complexity of the problem and its underlying structure: we prove that for k⩽3 the minimum number of necessary recolourings is polynomial in the size of the graph, while for k⩾4 instances exist where this number is superpolynomial.


Bonsma, P., Cereceda, L., van den Heuvel, J., & Johnson, M. (2007). Finding Paths between Graph Colourings: Computational Complexity and Possible Distances. Electronic Notes in Discrete Mathematics, 29, 463-469.

Journal Article Type Article
Publication Date Aug 15, 2007
Deposit Date Mar 17, 2014
Journal Electronic Notes in Discrete Mathematics
Print ISSN 1571-0653
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 29
Pages 463-469
Keywords Colour graph, PSPACE-completeness, Superpolynomial paths.