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Clause Learning and New Bounds for Graph Coloring

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 11008)


Graph coloring is a major component of numerous allocation and scheduling problems.

We introduce a hybrid CP/SAT approach to graph coloring based on exploring Zykov’s tree: for two non-neighbors, either they take a different color and there might as well be an edge between them, or they take the same color and we might as well merge them. Branching on whether two neighbors get the same color yields a symmetry-free tree with complete graphs as leaves, which correspond to colorings of the original graph.

We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; and a branching heuristic emulating Brelaz on the Zykov tree.

The combination of these techniques in both a branch-and-bound and in a bottom-up search outperforms Dsatur and other SAT-based approaches on standard benchmarks both for finding upper bounds and for proving lower bounds.

G. Katsirelos—Partially supported by the french “Agence nationale de la Recherche”, project DEMOGRAPH, reference ANR-16-C40-0028.

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  1. 1.

    We abuse the neighborhood notation and write \(N({S})\) for \(\bigcup _{u \in S}N({u})\).

  2. 2.

    We assume the graph is connected, otherwise \(u \) may not always exist.

  3. 3.

    Sources available at:

  4. 4.

    Sources available at:

  5. 5.

    cdcl exceeded the memory limit on 4 instances, and CPLEX on 16 instances.


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Correspondence to Emmanuel Hebrard or George Katsirelos .

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Hebrard, E., Katsirelos, G. (2018). Clause Learning and New Bounds for Graph Coloring. In: Hooker, J. (eds) Principles and Practice of Constraint Programming. CP 2018. Lecture Notes in Computer Science(), vol 11008. Springer, Cham.

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