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Graphs and Combinatorics

, Volume 35, Issue 1, pp 1–31 | Cite as

Polynomial \(\chi \)-Binding Functions and Forbidden Induced Subgraphs: A Survey

  • Ingo Schiermeyer
  • Bert Randerath
Original Paper
  • 53 Downloads

Abstract

A graph G with clique number \(\omega (G)\) and chromatic number \(\chi (G)\) is perfect if \(\chi (H)=\omega (H)\) for every induced subgraph H of G. A family \({\mathcal {G}}\) of graphs is called \(\chi \)-bounded with binding function f if \(\chi (G') \le f(\omega (G'))\) holds whenever \(G \in {\mathcal {G}}\) and \(G'\) is an induced subgraph of G. In this paper we will present a survey on polynomial \(\chi \)-binding functions. Especially we will address perfect graphs, hereditary graphs satisfying the Vizing bound (\(\chi \le \omega +1\)), graphs having linear \(\chi \)-binding functions and graphs having non-linear polynomial \(\chi \)-binding functions. Thereby we also survey polynomial \(\chi \)-binding functions for several graph classes defined in terms of forbidden induced subgraphs, among them \(2K_2\)-free graphs, \(P_k\)-free graphs, claw-free graphs, and \({ diamond}\)-free graphs.

Families of\(\chi \)-bound graphs are natural candidates for polynomial approximation algorithms for the vertex coloring problem. (András Gyárfás [42])

Keywords

Chromatic number Perfect graphs \(\chi \)-bounded \(\chi \)-binding function Forbidden induced subgraph 

Notes

Acknowledgements

We thank an anonymous reviewer for several valuable comments.

References

  1. 1.
    Addario-Berry, L., Chudnovsky, M., Havet, F., Reed, B., Seymour, P.: Bisimplicial vertices in even-hole-free graphs. J. Combin. Theory B 98, 1119–1164 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atminas, A., Lozin, V., Zamaraev, V.: Linear Ramsey Numbers. Lecture Notes in Computer Science, pp. 26–38 (2018)Google Scholar
  3. 3.
    Bacsó, G., Tuza, Zs: Dominating cliques in \(P_5\)-free graphs. Period. Math. Hungar. 21(3), 303–308 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beineke, L.W.: Derived graphs and digraphs. In: Sachs, H. (ed.) Beiträge zur Graphentheorie, pp. 17–33. Teubner, Leipzig (1968)Google Scholar
  5. 5.
    Berge, C.: Perfect Graphs, Six Papers on Graph Theory, pp. 1–21. Indian Statistical Institute, Calcutta (1963)Google Scholar
  6. 6.
    Bharathi, A., Choudum, S.A.: Colouring of \((P_3 \cup P_2)\)-free graphs. Graphs Combin. 34(1), 97–107 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blázsik, Z., Hujter, M., Pluhár, A., Tuza, Zs: Graphs with no induced \(C_4\). Discrete Math. 115, 51–55 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bollobás, B.: The independence ratio of regular graphs. Proc. Am. Math. Soc. 83(2), 433–436 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  10. 10.
    Brandt, S.: Triangle-free graphs and forbidden subgraphs. Discrete Math. 120, 25–33 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brause, C., Doan, T.D., Schiermeyer, I.: On the chromatic number of \(P_5, K_{2, t}\) graphs. Electron. Notes Discrete Math. 55, 127–130 (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Brause, C., Holub, P., Kabela, A., Ryjáček, Z., Schiermeyer, I., Vrána, P.: On forbidden subgraphs for \(K_{1,3}\)-free perfect graphs (2018) (preprint, submitted) Google Scholar
  13. 13.
    Brause, C., Randerath, B., Schiermeyer, I., Vumar, E.: On the chromatic number of \(2K_2\)-free graphs. Discrete Appl. Math.  https://doi.org/10.1016/j.dam.2018.09.030 [Extended abstract in: Bordeaux Graph Workshop, pp. 50–53 (2016)]
  14. 14.
    Cameron, K., Chaplick, S., Hoáng, C.T.: On the structure of \((pan, even hole)\)-free graphs. J. Graph Theory 87(1), 108–129 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cameron, K., Huang, S., Merkel, O.: An optimal \(\chi \)-bound for \((P_6,diamond)\)-free graphs (2018) (preprint). arXiv:1809.00739 v1
  16. 16.
    Cameron, K., da Silva, M., Huang, S., Vušković, K.: Structure and algorithms for \((cap, even hole)\)-free graphs. Discrete Math. 341(2), 463–473 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Choudum, S., Karthick, T.: Maximal cliques in \((P_2 \cup P_3, C_4)\)-free graphs. Discrete Math. 310, 3398–3403 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Choudum, S., Karthick, T., Shalu, M.: Linear chromatic bounds for a subfamily of \(3K_1\)-free graphs. Graphs Combin. 24, 413–428 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Choudum, S., Karthick, T., Shalu, M.: Perfect coloring and linearly \(\chi \)-bound \(P_6\)-free graphs. J. Graph Theory 54, 293–306 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chudnovsky, M.: The Erdős–Hajnal conjecture—a survey. J. Graph Theory 75, 178–190 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chudnovsky, M., Seymour, P.: Claw-free graphs VI. Colouring. J. Combin. Theory B 100(6), 560–572 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chudnovsky, M., Sivaraman, V.: Perfect divisibility and 2-divisibility. J. Graph Theory (2018).  https://doi.org/10.1002/jgt.22367
  23. 23.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: \(K_4\)-free graphs with no odd holes. J. Combin. Theory B 100, 313–331 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Chudnovsky, M., Penev, I., Scott, A., Trotignon, N.: Substitution and \(\chi \)-boundness. J. Combin. Theory B 103, 567–586 (2013)CrossRefzbMATHGoogle Scholar
  26. 26.
    Chudnovsky, M., Esperet, L., Lemoine, L., Maceli, P., Maffray, F., Penev, I.: Graphs with no induced five-vertex path or antipath. J. Graph Theory 84, 221–232 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Chudnovsky, M., Scott, A., Seymour, P.: Induced subgraphs of graphs with large chromatic number. III. Long holes. Combinatorica 37(6), 1057–1072 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Chudnovsky, M., Scott, A., Seymour, P.: Induced subgraphs of graphs with large chromatic number. XII. Distant stars, (submitted) Google Scholar
  29. 29.
    Chudnovsky, M., Scott, A., Seymour, P., Spirkl, S.: Induced subgraphs of graphs with large chromatic number. VIII. Long odd holes (submitted) Google Scholar
  30. 30.
    Chung, F.R.K.: On the covering of graphs. Discrete Math. 30, 89–93 (1980)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Chvátal, V., Sbihi, N.: Recognizing claw-free perfect graphs. J. Combin. Theory B 44, 154–176 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Conforti, M., Cornuéjols, G., Kapoor, A., Vušković, K.: Even and odd holes in cap-free graphs. J. Graph Theory 30, 289–308 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Dhanalakshmi, S., Sadagopan, N., Manogna, V.: On \(2K_2\)-free graphs—structural and combinatorial view. arXiv:1602.03802v2 [math.CO] (2016)
  34. 34.
    Erdős, P.: Graph theory and probability. Can. J. Math. 11, 34–38 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Esperet, L., Lemoine, L., Maffray, F., Morel, G.: The chromatic number of \(\{P_5, K_4\}\)-free graphs. Discrete Math. 313, 743–754 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Fan, G., Xu, B., Ye, T., Yu, X.: Forbidden subgraphs and 3-colorings. SIAM J. Discrete Math. 28, 1226–1256 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Fouquet, J.L., Giakoumakis, V., Maire, F., Thuillier, H.: On graphs without \(P_5\) and \({\bar{P}}_5\). Discrete Math. 146, 33–44 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Fraser, D., Hamel, A., Hoàng, C.: On the structure of \((even-hole, kite)\)-free graphs. Graphs Combin. 34, 989–99 (2018)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gaspers, S., Huang, S.: \((2P_2,K_4)\)-free graphs are 4-colorable. arXiv:1807.05547v1 [math.CO] (2018)
  40. 40.
    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 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Gravier, S., Hoàng, C.T., Maffray, F.: Coloring the hypergraph of maximalcliques of a graph with no long path. Discrete Math. 272, 283–290 (2003)CrossRefzbMATHGoogle Scholar
  42. 42.
    Gyárfás, A.: Problems from the world surrounding perfect graphs. In Proc. Int. Conf. on Comb. Analysis and Applications (Pokrzywna, 1985). Zastos. Mat. 19, 413–441 (1987)MathSciNetGoogle Scholar
  43. 43.
    Gyárfás, A., Szemerédi, E., Tuza, Zs: Induced subtrees in graphs of large chromatic number. Discrete Math. 30, 235–244 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Henning, M., Löwenstein, C., Rautenbach, D.: Independent sets and matchings in subcubic graphs. Discrete Math. 312, 1900–1910 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Hoàng, C.: On the structure of (banner, odd hole)-free graphs. J. Graph Theory.  https://doi.org/10.1002/jgt.22258
  46. 46.
    Hoàng, C., McDiarmid, C.: On the divisibility of graphs. Discrete Math. 242, 145–156 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Hougardy, S.: Classes of perfect graphs. Discrete Math. 306, 2529–2571 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Javdekar, M.: Note on Choudom’s chromatic bound for a class of graphs. J. Graph Theory 4, 265–268 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Jensen, T.R., Toft, B.: Graph Colouring Problems. Wiley, New York (1995)zbMATHGoogle Scholar
  50. 50.
    Karthick, T., Maffray, F.: Vizing bound for the chromatic number on some graph classes. Graphs Combin. 32, 1447–1460 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Karthick, T., Maffray, F.: Square-free graphs with no six-vertex induced path (preprint) (2018). arXiv:1805.05007v1 [cs.DM]
  52. 52.
    Karthick, T., Maffray, F.: Coloring (gem, co-gem)-free graphs. J. Graph Theory 89, 288–303 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Karthick, T., Mishra, S.: Chromatic bounds for some classes of \(2K_2\)-free graphs. Discrete Math. 341(11), 3079–3088 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Karthick, T., Mishra, S.: On the chromatic number of \((P_6, diamond)\)-free graphs. Graphs Combin. 34, 677–692 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Kierstead, H.: On the chromtaic index of multigraphs without large triangles. J. Combin. Theory B 36, 156–160 (1984)CrossRefzbMATHGoogle Scholar
  56. 56.
    Kierstead, H., Penrice, S.: Radius two trees specify \(\chi \)-bounded classes. J. Graph Theory 18, 119–129 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Kierstead, H., Zhu, Y.: Radius three trees in graphs with large chromatic number. SIAM J. Discrete Math. 17, 571–581 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Kim, J.H.: The Ramsey number \(R(3, t)\) has order of magnitude \(t^2/log t\). Random Struct. Algorithms 7, 173–207 (1995)CrossRefzbMATHGoogle Scholar
  59. 59.
    Kloks, T., Müller, H., Vušković, K.: Even-hole free graphs that do not contain diamonds: a structure theorem and its consequences. J. Combin. Theory B 99, 733–800 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Lovász, L.: A characterization of perfect graphs. J. Combin. Theory B 13(2), 95–98 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press, Baltimore (2001)zbMATHGoogle Scholar
  62. 62.
    Molloy, M., Reed, B.: Graph Colourings and the Probabilistic Method. Algorithms Comb., vol. 23. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  63. 63.
    Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Olariu, S.: Paw-free graphs. Inf. Process. Lett. 28, 53–54 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Ramirez-Alfonsin, J.L., Reed, B.: Perfect Graphs. Springer, Berlin (2001)zbMATHGoogle Scholar
  66. 66.
    Randerath, B.: The Vizing bound for the chromatic number based on forbidden pairs, Ph.D. thesis. RWTH Aachen, Shaker Verlag (1998)Google Scholar
  67. 67.
    Randerath, B.: \(3\)-Colorability and forbidden subgraphs. I: chracterizing pairs. Discrete Math. 276, 313–325 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Randerath, B., Schiermeyer, I., Tewes, M.: \(3\)-Colorability and forbidden subgraphs. II: polynomial algorithms. Discrete Math. 251(1–3), 137–153 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Randerath, B., Schiermeyer, I.: Vertex colouring and forbidden subgraphs—a survey. Graphs Combin. 20(1), 1–40 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Roussel, F., Rusu, I., Thuillier, H.: The Strong Perfect Graph Conjecture: 40 years of attempts, and its resolution. Discrete Math. 309, 6092–6113 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Schiermeyer, I.: Chromatic number of \(P_5\)-free graphs: Reed’s conjecture. Discrete Math. 339(7), 1940–1943 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Schiermeyer, I.: On the chromatic number of \((P_5, windmill)\)-free graphs. Opusc. Math. 37(4), 609–615 (2017)CrossRefzbMATHGoogle Scholar
  73. 73.
    Scott, A.D.: Induced trees in graphs of large chromatic number. J. Graph Theory 24, 297–311 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Scott, A., Seymour, P.: Induced subgraphs of graphs with large chromatic number. I. Odd holes. J. Combin. Theory B 121, 68–84 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Scott, A., Seymour, P.: Induced subgraphs of graphs with large chromatic number. XIII. New Brooms (submitted) Google Scholar
  76. 76.
    Sivaraman, V.: Some problems on induced subgraphs. Discrete Appl. Math. 236, 422–427 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Sumner, D.P.: Subtrees of a graph and the chromatic number. In: Chartrand, G. (ed.) The Theory and Applications of Graphs, 4th Int. Conf., Kalamazoo/Mich., pp. 557–576. Wiley, New York (1980)Google Scholar
  78. 78.
    Truemper, K.: Alpha-balanced graphs and matrices and GF(3)-representability of matroids. J. Combin. Theory B 32, 112–139 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Tuza, Z.: Graph colorings with local constraints—a survey. Discuss. Math. Graph Theory 17, 161–228 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Vušković, K.: Even-hole free graphs: a survey. Appl. Anal. Discrete Math. 4, 219–240 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Wagon, S.: A bound on the chromatic number of graphs without certain induced subgraphs. J. Combin. Theory B 29, 345–346 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Yuditsky, Y.: Workshop “New trends in graph colouring” at BIRS (private communication) (2016)Google Scholar

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© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und AlgebraTechnische Universität Bergakademie FreibergFreibergGermany
  2. 2.Institut für NachrichtentechnikTechnische Hochschule KölnCologneGermany

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