The computational complexity of weighted vertex coloring for \(\{P_5,K_{2,3},K^+_{2,3}\}\)-free graphs


In this paper, we show that the weighted vertex coloring problem can be solved in polynomial on the sum of vertex weights time for \(\{P_5,K_{2,3}, K^+_{2,3}\}\)-free graphs. As a corollary, this fact implies polynomial-time solvability of the unweighted vertex coloring problem for \(\{P_5,K_{2,3},K^+_{2,3}\}\)-free graphs. As usual, \(P_5\) and \(K_{2,3}\) stands, respectively, for the simple path on 5 vertices and for the biclique with the parts of 2 and 3 vertices, \(K^+_{2,3}\) denotes the graph, obtained from a \(K_{2,3}\) by joining its degree 3 vertices with an edge.

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  1. 1.

    Cameron, K., Huang, S., Penev, I., Sivaraman, V.: The class of \((P_7, C_4, C_5)\)- free graphs: decomposition, algorithms, and \(\chi \)-boundedness. J. Graph Theory (2019).

    Article  Google Scholar 

  2. 2.

    Cameron, K., da Silva, M., Huang, S., Vuskovic, K.: Structure and algorithms for \((cap, even~hole)\)-free graphs. Discrete Math. 341, 463–473 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Dabrowski, K., Dross, F., Paulusma, D.: Colouring diamond-free graphs. J. Comput. Syst. Sci. 89, 410–431 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Dabrowski, K., Golovach, P., Paulusma, D.: Colouring of graphs with Ramsey-type forbidden subgraphs. Theor. Comput. Sci. 522, 34–43 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dabrowski, K., Lozin, V., Raman, R., Ries, B.: Colouring vertices of triangle-free graphs without forests. Discrete Math. 312, 1372–1385 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dabrowski, K., Paulusma, D.: On colouring \((2P_2, H)\)-free and \((P_5, H)\)-free graphs. Inf. Process. Lett. 134, 35–41 (2018)

    Article  Google Scholar 

  7. 7.

    Dai, Y., Foley, A., Hoàng, C.: On coloring a class of claw-free graphs: to the memory of Frédéric Maffray. Electron. Notes Theor. Comput. Sci. 346, 369–377 (2019)

    Article  Google Scholar 

  8. 8.

    Foley, A., Fraser, D., Hoàng, C., Holmes, K., LaMantia, T.: The intersection of two vertex coloring problems. arXiv:1904.08180

  9. 9.

    Fraser, D., Hamela, A., Hoàng, C., Holmes, K., LaMantia, T.: Characterizations of \((4K_1, C_4, C_5)\)-free graphs. Discrete Appl. Math. 231, 166–174 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gaspers, S., Huang, S., Paulusma, D.: Colouring square-free graphs without long induced paths. J. Comput. Syst. Sci. 106, 60–79 (2019)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Golovach, P., Johnson, M., Paulusma, D., Song, J.: A survey on the computational complexity of coloring graphs with forbidden subgraphs. J. Graph Theory 84, 331–363 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Golovach, P., Paulusma, D., Ries, B.: Coloring graphs characterized by a forbidden subgraph. Discrete Appl. Math. 180, 101–110 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hell, P., Huang, S.: Complexity of coloring graphs without paths and cycles. Discrete Appl. Math. 216, 211–232 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hoàng, C., Lazzarato, D.: Polynomial-time algorithms for minimum weighted colorings of \((P_5,\overline{P_5})\)-free graphs and similar graph classes. Discrete Appl. Math. 186, 106–111 (2015)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Karthick, T., Maffray, F., Pastor, L.: Polynomial cases for the vertex coloring problem. Algorithmica 81, 1053–1074 (2017)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.: Complexity of coloring graphs without forbidden induced subgraphs. Lect. Notes Comput. Sci. 2204, 254–262 (2001)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kim, D., Du, D.-Z., Pardalos, P.M.: A coloring problem on the \(n\)-cube. Discrete Appl. Math. 103, 307–311 (2000)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Lozin, V., Malyshev, D.: Vertex coloring of graphs with few obstructions. Discrete Appl. Math. 216, 273–280 (2017)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Malyshev, D.: The coloring problem for classes with two small obstructions. Optim. Lett. 8, 2261–2270 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Malyshev, D.: Two cases of polynomial-time solvability for the coloring problem. J. Comb. Optim. 31, 833–845 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Malyshev, D.: Polynomial-time approximation algorithms for the coloring problem in some cases. J. Comb. Optim. 33, 809–813 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Malyshev, D.: The weighted coloring problem for two graph classes characterized by small forbidden induced structures. Discrete Appl. Math. 247, 423–432 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Malyshev, D., Lobanova, O.: Two complexity results for the vertex coloring problem. Discrete Appl. Math. 219, 158–166 (2017)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Pardalos, P.M., Mavridou, T., Xue, J.: The graph coloring problem: a bibliographic survey. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 1077–1141. Springer, Boston (1998)

    Google Scholar 

  25. 25.

    Wang, H., Pardalos, P.M., Liu, B.: Optimal channel assignment with list-edge coloring. J. Combin. Optim. 38, 197–207 (2019)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Wang, H., Wu, L., Wu, W., Pardalos, P.M., Wu, J.: Minimum total coloring of planar graph. J. Glob. Optim. 60, 777–791 (2014)

    MathSciNet  Article  Google Scholar 

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Research is supported under financial support of Russian Science Foundation, project No 19-71-00005.

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Correspondence to D. S. Malyshev.

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Malyshev, D.S., Razvenskaya, O.O. & Pardalos, P.M. The computational complexity of weighted vertex coloring for \(\{P_5,K_{2,3},K^+_{2,3}\}\)-free graphs. Optim Lett 15, 137–152 (2021).

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  • Coloring problem
  • Hereditary class
  • Computational complexity