Abstract
We consider a weighted version of the subcoloring problem that we call the hypocoloring problem: given a weighted graph G=(V,E;w) where w(v)≥ 0, the goal consists in finding a partition \({\cal S}=(S_1,\ldots,S_k)\) of the node set of G into hypostable sets and minimizing ∑\(_{i=1}^{k}\) w(S i ) where an hypostable S is a subset of nodes which generates a collection of node disjoint cliques K. The weight of S is defined as max { ∑ v ∈ K w(v)| K ∈ S}. Properties of hypocolorings are stated; complexity and approximability results are presented in some graph classes. The associated decision problem is shown to be NP-complete for bipartite graphs and triangle-free planar graphs with maximum degree 3. Polynomial algorithms are given for graphs with maximum degree 2 and for trees with maximum degree Δ.
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de Werra, D., Demange, M., Monnot, J., Paschos, V.T. (2004). The Hypocoloring Problem: Complexity and Approximability Results when the Chromatic Number Is Small. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2004. Lecture Notes in Computer Science, vol 3353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30559-0_32
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DOI: https://doi.org/10.1007/978-3-540-30559-0_32
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