Using Hajós’ Construction to Generate Hard Graph 3-Colorability Instances

  • Sheng Liu
  • Jian Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4120)


In this paper we propose a constructive algorithm using constraint propagation to generate 4-critical graph units (4-CGUs) which have only one triangle as subgraph. Based on these units we construct 4-critical graphs using Hajós’ join construction. By choosing Grotztsch graph as the initial graph and carefully selecting the edge to be joined, we make sure that the generated graphs are 4-critical and triangle-free. Experiments show that these graphs are exceptionally hard for backtracking algorithms adopting Brélaz’s heuristics. We also give some preliminary analysis on the source of hardness.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sheng Liu
    • 1
    • 2
  • Jian Zhang
    • 1
  1. 1.Laboratory of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingChina
  2. 2.Graduate UniversityChinese Academy of SciencesBeijingChina

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