Abstract
In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erdős and independently Albertson et al. proved that every graph of maximum degree at most 3 has a 2-subcoloring.We point out in this paper that this fact is best possible with respect to degree-constraints by showing that the problem of recognizing 2-subcolorable graphs with maximum degree 4 is NP-complete, even when restricted to triangle-free planar graphs. Moreover, in general, for fixed k, recognizing k-subcolorable graphs is NP-complete on graphs with maximum degree at most k 2. In contrast, we show that, for arbitrary k, k-SUBCOLORABILITY can be computed efficiently on graphs of bounded treewidth and on cographs.
Research supported in part by EU ARACNE project HPRN-CT-1999-00112 and EU APPOL project IST-1999-14084
Financial support by Deutsche Forschungsgemeinschaft is gratefully acknowledged.
Supported by the Ministery of Education of the Czech Republic as project LN00A056
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References
D. Achlioptas, The complexity of G-free graph colourability, Discrete Math, 165/166 (1997), 31–38.
M.O. Albertson, R.E. Jamison, S.T. Hedetniemi, S.C. Locke, The subchromatic number of a graph, Discrete Math., 74 (1989), 33–49.
H.L. Bodlaender, A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Computing, 25 (1996), 1305–1317.
I. Broere, C.M. Mynhardt, Generalized colorings of outerplanar and planar graphs, Proc. Graph Theory with Applications to Algorithms and Computer Science, Kalamazoo, Mich., (1984), 151–161.
J.L. Brown, D.G. Corneil, On generalized graph colorings, J. Graph Theory, 11 (1987), 87–99.
D.G. Corneil, H. Lerchs, L. Stewart Burlingham, Complement reducible graphs. Discrete Appl. Math., 3 (1981), 163–174.
D.G. Corneil, Y. Perl, L.K. Stewart, A linear recognition algorithm for cographs, SIAM J. Computing, 14 (1985), 926–934.
B. Courcelle, Graph rewriting: an algebraic and logical approach, Handbook of Theoretical Computer Science, volume B, (1990), 192–242.
B. Courcelle, The monadic second-order logic of graphs I: Recognizable sets of finite graphs, Information and Computation, 85 (1991), 12–75.
P. Erdős, Bipartite subgraphs of graphs, Math. Lapok, 18 (1967), 283–288.
M. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W.H. Freeman, San Fransisco, 1979.
M.R. Garey, D.S. Johnson, L. Stockmayer, Some simplified NP-complete graph problems, Theoretical Computer Science, 1 (1976), 237–267.
T. Kloks, Treewidth — Computations and Approximations, Lecture Notes in Computer Science no. 842, Springer-Verlag, 1994.
F. Mafray, M. Preissman, On the NP-completeness of k-colorability problem of triangle-free graphs, Discrete Math., 162 (1996), 313–317.
C.M. Mynhardt, I. Broere, Generalized colorings of graphs, Proc. Graph Theory with Applications to Algorithms and Computer Science, Kalamazoo, Mich., (1984), 583–594.
N. Robertson and P.D. Seymour, Graph minors. II. Algorithmic aspects of treewidth, Journal of Algorithms, 7 (1986), 309–322.
T.J. Schaefer, The complexity of the satisfability problem, Proc. 10th Ann. ACM Symp. on theory of computing (1978), 216–226.
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Fiala, J., Jansen, K., Le, V.B., Seidel, E. (2001). Graph Subcolorings: Complexity and Algorithms. In: Brandstädt, A., Le, V.B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2001. Lecture Notes in Computer Science, vol 2204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45477-2_15
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DOI: https://doi.org/10.1007/3-540-45477-2_15
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