, Volume 39, Issue 2, pp 377–410 | Cite as

Defective Colouring of Graphs Excluding A Subgraph or Minor

  • Patrice Ossona De MendezEmail author
  • Sang-Il Oum
  • David R. Wood


Archdeacon (1987) proved that graphs embeddable on a fixed surface can be 3-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no Kt+1-minor can be t-coloured so that each colour class induces a subgraph of bounded maximum degree. We prove a common generalisation of these theorems with a weaker assumption about excluded subgraphs. This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with relevance to the earth-moon problem), graphs with given stack- or queue-number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, and graphs excluding a complete bipartite graph as a topological minor.

Mathematics Subject Classification (2010)

05C15 05C83 05C10 


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  1. [1]
    B. Albar and D. Goncàlves: On triangles in Kr-minor free graphs, J. Graph Theory 88 (2018), 154–173.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. O. Albertson, D. L. Boutin and E. Gethner: More results on r-in ated graphs: arboricity, thickness, chromatic number and fractional chromatic number, Ars Math. Contemp. 4 (2011), 5–24.zbMATHGoogle Scholar
  3. [3]
    D. Archdeacon: A note on defective colorings of graphs in surfaces, J. Graph Theory 11 (1987), 517–519.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Barát, G. Joret and D. R. Wood: Disproof of the list Hadwiger conjecture, Electron. J. Combin. 18, 2011.
  5. [5]
    F. Bernhart and P. C. Kainen: The book thickness of a graph, J. Combin. Theory Ser. B 27 (1979), 320–331.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Blain, G. Bowlin, T. Fleming, J. Foisy, J. Hendricks and J. Lacombe: Some results on intrinsically knotted graphs, J. Knot Theory Ramifications 16 (2007), 749–760.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    B. Bollobás and A. Thomason: Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, European J. Combin. 19 (1998), 883–887.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    O. V. Borodin: On the total coloring of planar graphs, J. Reine Angew. Math. 394 (1989), 180–185.MathSciNetzbMATHGoogle Scholar
  9. [9]
    O. V. Borodin and A. V. Kostochka: Defective 2-colorings of sparse graphs, J. Combin. Theory Ser. B 104 (2014), 72–80.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    O. V. Borodin and D. P. Sanders: On light edges and triangles in planar graphs of minimum degree five, Math. Nachr. 170 (1994), 19–24.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    O. V. Borodin, A. V. Kostochka, N. N. Sheikh and G. Yu: M-degrees of quadrangle-free planar graphs, J. Graph Theory 60 (2009), 80–85.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    O. V. Borodin, A. O. Ivanova, M. Montassier and A. Raspaud: (k, j)-coloring of sparse graphs, Discrete Appl. Math. 159 (2011), 1947–1953.CrossRefzbMATHGoogle Scholar
  13. [13]
    O. V. Borodin, A. O. Ivanova, M. Montassier and A. Raspaud: (k, 1)-coloring of sparse graphs, Discrete Math. 312 (2012), 1128–1135.CrossRefzbMATHGoogle Scholar
  14. [14]
    O. V. Borodin, A. Kostochka and M. Yancey: On 1-improper 2-coloring of sparse graphs, Discrete Math. 313 (2013), 2638–2649.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    P. Bose, M. Smid and D. R. Wood: Light edges in degree-constrained graphs, Discrete Math. 282 (2004), 35–41.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Chen and A. Raspaud: On (3, 1)*-choosability of planar graphs without adjacent short cycles, Discrete Appl. Math. 162 (2014), 159–166.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Chen, A. Raspaud and W. Wang: A (3,1)*-choosable theorem on planar graphs, J. Comb. Optim. 32 (2016), 927–940.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    I. Choi and L. Esperet: Improper coloring of graphs on surfaces, arXiv: 1603.02841, 2016.Google Scholar
  19. [19]
    I. Choi and A. Raspaud: Planar graphs with girth at least 5 are (3, 5)-colorable, Discrete Math. 338 (2015), 661–667.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Y. Colin de Verdière: Sur un nouvel invariant des graphes et un critère de planarit é, J. Combin. Theory Ser. B 50 (1990), 11–21.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Y. Colin de Verdière: On a new graph invariant and a criterion for planarity, in: Graph structure theory, volume 147 of Contemp. Math., 137–147, Amer. Math. Soc., 1993.CrossRefGoogle Scholar
  22. [22]
    J. H. Conway and C. M. Gordon: Knots and links in spatial graphs, J. Graph Theory 7 (1983), 445–453.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    R. Corrêa, F. Havet and J.-S. Sereni: About a Brooks-type theorem for improper colouring, Australas. J. Combin. 43 (2009), 219–230.MathSciNetzbMATHGoogle Scholar
  24. [24]
    L. Cowen, W. Goddard and C. E. Jesurum: Defective coloring revisited, J. Graph Theory 24 (1997), 205–219.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    L. J. Cowen, R. H. Cowen and D. R. Woodall: Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J. Graph Theory 10 (1986), 187–195.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    E. de Klerk, D. V. Pasechnik and G. Salazar: Book drawings of complete bipartite graphs, Discrete Appl. Math. 167 (2014), 80–93.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    R. Diestel: Graph theory, volume 173 of Graduate Texts in Mathematics, Springer, 4th edition, 2010.CrossRefzbMATHGoogle Scholar
  28. [28]
    P. Dorbec, T. Kaiser, M. Montassier and A. Raspaud: Limits of near-coloring of sparse graphs, J. Graph Theory 75 (2014), 191–202.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    V. Dujmović and D. R. Wood: On linear layouts of graphs, Discrete Math. Theor. Comput. Sci. 6 (2004), 339–358.MathSciNetzbMATHGoogle Scholar
  30. [30]
    N. Eaton and T. Hull: Defective list colorings of planar graphs, Bull. Inst. Combin. Appl 25 (1999), 79–87.MathSciNetzbMATHGoogle Scholar
  31. [31]
    K. Edwards, D. Y. Kang, J. Kim, S. Oum and P. Seymour: A relative of Hadwiger's conjecture, SIAM J. Discrete Math. 29 (2015), 2385–2388.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    H. Enomoto, M. S. Miyauchi and K. Ota: Lower bounds for the number of edgecrossings over the spine in a topological book embedding of a graph, Discrete Appl. Math. 92 (1999), 149–155.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    J. Foisy: Intrinsically knotted graphs, J. Graph Theory 39 (2002), 178–187.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    E. Gethner and T. Sulanke: Thickness-two graphs, II. More new nine-critical graphs, independence ratio, cloned planar graphs, and singly and doubly outerplanar graphs, Graphs Combin. 25 (2009), 197–217.zbMATHGoogle Scholar
  35. [35]
    N. Goldberg, T. W. Mattman and R. Naimi: Many, many more intrinsically knotted graphs, Algebr. Geom. Topol. 14 (2014), 1801–1823.CrossRefzbMATHGoogle Scholar
  36. [36]
    D. J. Harvey and D. R. Wood: Average degree conditions forcing a minor, Electron. J. Combin. 23 (2016), #P1.42.Google Scholar
  37. [37]
    F. Havet and J.-S. Sereni: Improper choosability of graphs and maximum average degree, J. Graph Theory 52 (2006), 181–199.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang and X. Zhu: Edge-partitions of planar graphs and their game coloring numbers. J. Graph Theory 41 (2002), 307–317.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    L. S. Heath and A. L. Rosenberg: Laying out graphs using queues, SIAM J. Comput. 21 (1992), 927–958.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    K. Hendrey and D. R. Wood: The extremal function for Petersen minors, J. Combinatorial Theory Ser. B 131 (2018), 220–253.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J. P. Hutchinson: Coloring ordinary maps, maps of empires and maps of the moon, Math. Mag. 66 (1993), 211–226.zbMATHGoogle Scholar
  42. [42]
    J. Ivančo: The weight of a graph, Ann. Discrete Math. 51 (1992), 113–116.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    B. Jackson and G. Ringel: Variations on Ringel's earth-moon problem, Discrete Math. 211 (2000), 233–242.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    I. T. Jakobsen: Weakenings of the conjecture of Hadwiger for 8- and 9-chromatic graphs, Technical Report 22, Matematisk Institut, Aarhus Universitet, Denmark, 1971.Google Scholar
  45. [45]
    S. Jendrol' and T. Madaras: On light subgraphs in plane graphs of minimum degree five, Discuss. Math. Graph Theory 16 (1996), 207–217.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    S. Jendrol' and M. Tuhársky: A Kotzig type theorem for non-orientable surfaces, Mathematica Slovaca 56 (2006), 245–253.MathSciNetzbMATHGoogle Scholar
  47. [47]
    S. Jendrol' and H.-J. Voss: Light subgraphs of multigraphs on compact 2-dimensional manifolds, Discrete Math. 233 (2001), 329–351,.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    S. Jendrol' and H.-J. Voss: Light subgraphs of graphs embedded in 2-dimensional manifolds of Euler characteristic ≤0, A survey, in: Paul Erdős and his Mathematics, II, volume 11 of Bolyai Soc. Math. Stud., 375–411. János Bolyai Math. Soc., 2002.Google Scholar
  49. [49]
    C. D. Keys: Graphs critical for maximal bookthickness, Pi Mu Epsilon J. 6 (1975), 79–84.MathSciNetzbMATHGoogle Scholar
  50. [50]
    J. Kim, A. Kostochka and X. Zhu: Improper coloring of sparse graphs with a given girth, II: constructions, J. Graph Theory 81 (2016), 403–413.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    J. Komlós and E. Szemerédi: Topological cliques in graphs. II, Combin. Probab. Comput. 5 (1996), 79–90.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    A. V. Kostochka: The minimum Hadwiger number for graphs with a given mean degree of vertices, Metody Diskret. Analiz. 38 (1982), 37–58.MathSciNetzbMATHGoogle Scholar
  53. [53]
    A. V. Kostochka: Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica 4 (1984), 307–316.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    A. V. Kostochka and N. Prince: On Ks;t-minors in graphs with given average degree, Discrete Math. 308 (2008), 4435–4445.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    A. V. Kostochka and N. Prince: Dense graphs have K3;t minors, Discrete Math. 310 (2010), 2637–2654.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    A. V. Kostochka and N. Prince: On Ks;t-minors in graphs with given average degree, II, Discrete Math. 312 (2012), 3517–3522.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    D. Kühn and D. Osthus: Complete minors in Ks;s-free graphs, Combinatorica 25 (2005), 49–64.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    D. Kühn and D. Osthus: Forcing unbalanced complete bipartite minors, European J. Combin. 26 (2005), 75–81.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    K.-W. Lih, Z. Song, W. Wang and K. Zhang: A note on list improper coloring planar graphs, Appl. Math. Lett. 14 (2001), 269–273.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    W. Mader: Homomorphiesätze für Graphen, Math. Ann. 178 (1968), 154–168.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    P. Mutzel, T. Odenthal and M. Scharbrodt: The thickness of graphs: a survey, Graphs Combin. 14 (1998), 59–73.MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    J. Nešetřil and P. Ossona de Mendez: First order properties on nowhere dense structures, J. Symb. Log. 75 (2010), 868–887.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    J. Nešetřil and P. Ossona de Mendez: On nowhere dense graphs, European J. Combin. 32 (2011), 600–617.MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    J. Nešetřil and P. Ossona de Mendez: Sparsity (Graphs, Structures, and Algorithms), volume 28 of Algorithms and Combinatorics, Springer, 2012.zbMATHGoogle Scholar
  65. [65]
    J. Nešetřil, P. Ossona de Mendez and D. R. Wood: Characterisations and examples of graph classes with bounded expansion, European J. Combin. 33 (2011), 350–373.MathSciNetzbMATHGoogle Scholar
  66. [66]
    M. Ozawa and Y. Tsutsumi: Primitive spatial graphs and graph minors, Rev. Mat. Complut. 20 (2007), 391–406.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    J. Pach and G. Tóth: Graphs drawn with few crossings per edge, Combinatorica 17 (1997), 427–439.MathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    S. V. Pemmaraju: Exploring the Powers of Stacks and Queues via Graph Layouts, PhD thesis, Virginia Polytechnic Institute and State University, U.S.A., 1992.Google Scholar
  69. [69]
    J. L. Ramírez Alfonsín: Knots and links in spatial graphs: a survey, Discrete Math. 302 (2005), 225–242.MathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    B. Reed and D. R. Wood: Forcing a sparse minor, Combin. Probab. Comput. 25 (2016), 300–322.MathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    G. Ringel: Färbungsprobleme auf Flächen und Graphen, volume 2 of Mathematische Monographien, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959.zbMATHGoogle Scholar
  72. [72]
    G. Ringel: Das Geschlecht des vollständigen paaren Graphen, Abh. Math. Sem. Univ. Hamburg 28 (1965), 139–150.MathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    N. Robertson, P. D. Seymour and R. Thomas: Hadwiger's conjecture for K6-free graphs, Combinatorica 13 (1993), 279–361.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    N. Robertson, P. D. Seymour and R. Thomas: A survey of linkless embeddings, in: N. Robertson and P. D. Seymour, editors, Graph structure theory. Proc. of AMSIMS-SIAM Joint Summer Research Conf. on Graph Minors, volume 147 of Contempory Mathematics, 125–136, American Mathematical Society, 1993.Google Scholar
  75. [75]
    N. Robertson, P. Seymour and R. Thomas: Petersen family minors, J. Combin. Theory Ser. B 64 (1995), 155–184.MathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    H. Sachs: On a spatial analogue of Kuratowski's theorem on planar graphs — an open problem, in: M. Borowiecki, J. W. Kennedy, and M. M. Syslo, editors, Proc. Conf. on Graph Theory, volume 1018 of Lecture Notes in Mathematics, 230–241, Springer, 1983.CrossRefGoogle Scholar
  77. [77]
    A. Schrijver: Minor-monotone graph invariants, in: Surveys in combinatorics, volume 241 of London Math. Soc. Lecture Note Ser., 163–196, Cambridge Univ. Press, 1997.Google Scholar
  78. [78]
    F. Shahrokhi, L. A. Székely, O. Sýkora and I. Vrťo: Drawings of graphs on surfaces with few crossings, Algorithmica 16 (1996), 118–131.MathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    R. Škrekovski: List improper colorings of planar graphs with prescribed girth, Discrete Math. 214 (2000), 221–233.MathSciNetCrossRefzbMATHGoogle Scholar
  80. [80]
    R. Thomas and P. Wollan: An improved linear edge bound for graph linkages, European J. Combin. 26 (2005), 309–324.MathSciNetCrossRefzbMATHGoogle Scholar
  81. [81]
    A. Thomason: An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 261–265.MathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    A. Thomason: The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001), 318–338.MathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    H. van der Holst: On the graph parameters of Colin de Verdière, in: Ten years LNMB, 37–44, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997.Google Scholar
  84. [84]
    H. van der Holst, L. Lovász and A. Schrijver: The Colin de Verdière graph parameter, in: Graph theory and Combinatorial Biology, volume 7 of Bolyai Soc. Math. Stud., 29–85, János Bolyai Math. Soc., 1999.Google Scholar
  85. [85]
    Y. Wang and L. Xu: Improper choosability of planar graphs without 4-cycles, SIAM J. Discrete Math. 27 (2013), 2029–2037.MathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    D. R. Wood: Cliques in graphs excluding a complete graph minor, Electron. J. Combin. 23 (2016), #P3.18.Google Scholar
  87. [87]
    R. G. Wood and D. R. Woodall: Defective choosability of graphs without small minors. Electron. J. Combin. 16 (2009), #R92.Google Scholar
  88. [88]
    D. R. Woodall: Defective choosability of graphs in surfaces, Discuss. Math. Graph Theory 31 (2011), 441–459.MathSciNetCrossRefzbMATHGoogle Scholar
  89. [89]
    M. Yancey: Thickness for improper colorings, 2012, Scholar
  90. [90]
    H. Zhang: On (4, 1)*-choosability of toroidal graphs without chordal 7-cycles and adjacent 4-cycles, Commentationes Mathematicae Universitatis Carolinae 54 (2013), 339–344.MathSciNetzbMATHGoogle Scholar
  91. [91]
    H. Zhang: (3, 1)*-choosability of graphs of nonnegative characteristic without intersecting short cycles, Proceedings–Mathematical Sciences 126 (2016), 159–165.MathSciNetzbMATHGoogle Scholar
  92. [92]
    L. Zhang: A (3, 1)*-choosable theorem on toroidal graphs, Discrete Appl. Math. 160 (2012), 332–338.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  • Patrice Ossona De Mendez
    • 1
    • 2
    Email author
  • Sang-Il Oum
    • 3
  • David R. Wood
    • 4
  1. 1.Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557)ParisFrance
  2. 2.Computer Science Institute of CharlesUniversity (IUUK)Praha 1Czech Republic
  3. 3.Department of Mathematical SciencesKAISTDaejeonSouth Korea
  4. 4.School of Mathematical Sciences MonashUniversity MelbourneMelbourneAustralia

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