Abstract
Motivated by a problem in Scheduling Theory, we introduce the χt-coloring problem, a generalization of the chromatic number problem that places a bound of t on the size of any color class. For fixed t, we show that the perfect χt-coloring problem (in which each color class must have cardinality exactly t) can be expressed in the counting monadic second-order logic and, hence, has a linear-time algorithm over the class of graphs G of bounded treewidth: A solution is a partition of G into induced subgraphs, each isomorphic to a fixed graph consisting of t isolated vertices. The logical formalism generalizes to allow these t vertices to be t isomorphic connected components. The linear-time algorithm so derived for the perfect χt-coloring problem is used to design a linear-time algorithm for the optimization version of the general χt-coloring problem (for fixed t) on graphs of bounded treewidth. We also show that this problem has a polynomial-time algorithm on bipartite graphs.
Research supported by the Natural Sciences and Engineering Research Council of Canada.
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© 1995 Springer-Verlag Berlin Heidelberg
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Kaller, D., Gupta, A., Shermer, T. (1995). The χt-coloring problem. In: Mayr, E.W., Puech, C. (eds) STACS 95. STACS 1995. Lecture Notes in Computer Science, vol 900. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59042-0_92
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DOI: https://doi.org/10.1007/3-540-59042-0_92
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