The subchromatic number of a graph
Discrete Mathematics - Graph colouring and variations
On the NP-completeness of the k-colorability problem for triangle-free graphs
Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Planar Graph Coloring with Forbidden Subgraphs: Why Trees and Paths Are Dangerous
SWAT '02 Proceedings of the 8th Scandinavian Workshop on Algorithm Theory
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A partition of the vertices of a graph G into k pairwise disjoint sets V1, ..., Vk is called an (r1, ..., rk)-subcoloring if the subgraph Gi of G induced by Vi, 1 ≤ i ≤ k consists of disjoint complete subgraphs, each of which has cardinality no more than ri. Due to Erdös and Albertson et al., independently, every cubic (i.e., 3-regular) graph has a (2, 2)-subcoloring. Albertson et al. then asked for cubic graphs having (1,2)-subcolorings. We point out in this paper that this question is algorithmically difficult by showing that recognizing (1, 2)-subcolorable cubic graphs is NP-complete, even when restricted to triangle-free planar graphs.