Graphs and Combinatorics

, Volume 35, Issue 1, pp 239–248 | Cite as

Cut-Colorings in Coloring Graphs

  • Prateek Bhakta
  • Benjamin Brett Buckner
  • Lauren Farquhar
  • Vikram Kamat
  • Sara Krehbiel
  • Heather M. RussellEmail author
Original Paper


This paper studies the connectivity and biconnectivity of coloring graphs. For \(k\in \mathbb {N}\), the k-coloring graph of a base graph G has vertex set consisting of the proper k-colorings of G and edge set consisting of the pairs of k-colorings that differ on a single vertex. A base graph whose k-coloring graph is connected is said to be k-mixing; it is possible to transition between any two k-colorings in a k-mixing graph via a sequence of single vertex recolorings, where each intermediate step is also a proper k-coloring. A natural extension of connectedness is biconnectedness. If a base graph has a coloring graph that is not biconnected, then there exists a proper k-coloring that would disconnect the coloring graph if removed. We call such a coloring a k-cut coloring. We prove that no base graph that is 3-mixing can have a 3-cut coloring, but for any \(k\ge 4\) there exists a base graph that is k-mixing and has a k-cut coloring.


Graph coloring Reconfiguration Cut-coloring 



The authors acknowledge University of Richmond for continued support of this work.


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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Prateek Bhakta
    • 1
  • Benjamin Brett Buckner
    • 2
  • Lauren Farquhar
    • 3
  • Vikram Kamat
    • 4
  • Sara Krehbiel
    • 1
  • Heather M. Russell
    • 1
    Email author
  1. 1.University of RichmondRichmondUSA
  2. 2.Rensselaer Polytechnic InstituteTroyUSA
  3. 3.University of Colorado BoulderBoulderUSA
  4. 4.Villanova UniversityVillanovaUSA

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