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Coloring of Pseudocubic Graphs in Three Colors

  • S. N. SeleznevaEmail author
  • M. V. Mel’nikEmail author
  • A. V. AstakhovaEmail author
Article

Abstract

A graph is called pseudocubic if the degrees of all its vertices, with a single exception, do not exceed three, and the degree of an exceptional vertex does not exceed four. In this work, it is proved that the vertices of a pseudocubic graph without induced subgraphs that are isomorphic to K4 or K 4 can be colored in three colors. In addition, it is shown that the problem of 3-coloring of pseudocubic graphs can be solved using a polynomial algorithm.

Keywords

graph cubic graph subcubic graph pseudocubic graph degree of vertex coloring of vertices chromatic number 3-coloring of graphs polynomial algorithm 

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References

  1. 1.
    L. J. Stockmeyer, “Planar 3-colorability is NP-complete,” SIGACT News 5(3), 19–25 (1973).CrossRefGoogle Scholar
  2. 2.
    M. R. Garey, D. S. Johnson, and L. Stockmeyer, “Some simplified NP-complete graph problems,” Theor. Comput. Sci. 1, 237–267 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Konig, Theorie der endlichen und unendlichen Graphen (Leipzig, 1936).Google Scholar
  4. 4.
    H. Broersma, P. A. Golovach, D. Paulusma, and J. Song, “Updating the complexity status of coloring graphs without a fixed induced linear forest,” Theor. Comput. Sci. 414, 9–19 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P. Golovach, D. Paulusma, and J. Song, “4-coloring H-free graphs when H is small,” Discrete Appl. Math. 161, 140–150 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    C. T. Hoang, M. Kaminski, V. V. Lozin, J. Sawada, and X. Shu, “Deciding k-colorability of P 5-free graphs in polymomial time,” Algorithmica 57, 74–81 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. Bonomo, M. Chudnovsky, P. Maceli, O. Schaudt, M. Stein, and M. Zhong, “Three-coloringand list three-coloring of graphs without induced paths on seven vertices,” Combinatorica, 1–23 (2017).Google Scholar
  8. 8.
    S. Huang, “Improved complexity results on k-coloring P t-free graphs,” Eur. J. Combin. 51, 336–346 (2016).CrossRefzbMATHGoogle Scholar
  9. 9.
    R. L. Brooks, “On colouring the nodes of a network,” Proc. Cambridge Philos. Soc. 37, 194–197 (1941).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. A. Emelichev, O. I. Mel’nikov, V. I. Sarvanov, and R. I. Tyshkevich, Lectures on Graph Theory (Nauka, Moscow, 1990).zbMATHGoogle Scholar
  11. 11.
    J. A. Bondy and U. S. R. Murty, Graph Theory (Springer, London, 2008).CrossRefzbMATHGoogle Scholar
  12. 12.
    V. G. Vizing, “Coloring graph vertices in prescribed colors,” in Discrete Analysis Methods in the Theory of Codes and Schemes, Collection of Articles (Inst. Mat. SO AN SSSR, Novosibirsk, 1976), Vol. 29, pp. 3–10 [in Russian].Google Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Department of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia

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