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

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 Hajos' 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 Brelaz's heuristics. We also give some preliminary analysis on the source of hardness.

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