Finding Shortest Paths Between Graph Colourings

  • Matthew Johnson
  • Dieter KratschEmail author
  • Stefan Kratsch
  • Viresh Patel
  • Daniël Paulusma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)


The \(k\)-colouring reconfiguration problem asks whether, for a given graph \(G\), two proper \(k\)-colourings \(\alpha \) and \(\beta \) of \(G\), and a positive integer \(\ell \), there exists a sequence of at most \(\ell \) proper \(k\)-colourings of \(G\) which starts with \(\alpha \) and ends with \(\beta \) and where successive colourings in the sequence differ on exactly one vertex of \(G\). We give a complete picture of the parameterized complexity of the \(k\)-colouring reconfiguration problem for each fixed \(k\) when parameterized by \(\ell \). First we show that the \(k\)-colouring reconfiguration problem is polynomial-time solvable for \(k=3\), settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all \(k \ge 4\), we show that the \(k\)-colouring reconfiguration problem, when parameterized by \(\ell \), is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.



We are grateful to several reviewers for insightful comments that greatly improved our presentation.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Matthew Johnson
    • 1
  • Dieter Kratsch
    • 2
    Email author
  • Stefan Kratsch
    • 3
  • Viresh Patel
    • 4
  • Daniël Paulusma
    • 1
  1. 1.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  2. 2.LITAUniversité de LorraineMetz Cedex 01France
  3. 3.Technische Universität BerlinBerlinGermany
  4. 4.Queen Mary, University of LondonLondonUK

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