Abstract
A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. However, no efficient algorithm has been known for the total coloring problem on partial although a polynomial-time algorithm of very high order has been known. In this paper, we give a linear-time algorithm for the total coloring problem on partial k-trees with bounded k.
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References
S. Arnborg, B. Courcelle, A. Proskurowski and D. Seese, An algebraic theory of graph reduction, J. Assoc. Comput. Mach. 40(5), pp. 1134–1164, 1993.
S. Arnborg and J. Lagergren, Easy problems for tree-decomposable graphs, J. Algorithms, 12(2), pp. 308–340, 1991.
H.L. Bodlaender, Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees, Journal of Algorithms, 11(4), pp. 631–643, 1990.
R. B. Borie, R. G. Parker and C. A. Tovey, Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families, Algorithmica, 7, pp. 555–581, 1992.
B. Courcelle, The monadic second-order logic of grpahs I: Recognizable sets of finite graphs, Inform. Comput., 85, pp. 12–75, 1990.
B. Courcelle and M. Mosbath, Monadic second-order evaluations on tree-decomposable graphs, Theoret. Comput. Sci., 109, pp.49–82, 1993.
S. Isobe, X. Zhou and T. Nishizeki, A polynomial-time algorithm for finding total colorings of partial k-trees, Int. J. Found. Comput. Sci., 10(2), pp. 171–194, 1999.
A. Sánchez-Arroyo. Determining the total colouring number is NP-hard, Discrete Math., 78, pp. 315–319, 1989.
X. Zhou, S. Nakano and T. Nishizeki, Edge-coloring partial k-trees, J. Algorithms, 21, pp. 598–617, 1996.
X. Zhou, H. Suzuki and T. Nishizeki, A linear algorithm for edge-coloring series-parallel multigraphs, J. Algorithm, 20, pp. 174–201, 1996.
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© 1999 Springer-Verlag Berlin Heidelberg
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Isobe, S., Zhou, X., Nishizeki, T. (1999). A Linear Algorithm for Finding Total Colorings of Partial k-Trees . In: Algorithms and Computation. ISAAC 1999. Lecture Notes in Computer Science, vol 1741. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46632-0_35
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DOI: https://doi.org/10.1007/3-540-46632-0_35
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