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Chromatic Polynomials and Chromaticity of Graphs

Profile image of Fengming DongFengming Dong

2005

https://doi.org/10.1142/9789812569462
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Abstract
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This research explores chromatic polynomials and the chromaticity of graphs, addressing their historical significance and development in tackling problems such as the four-colour theorem. Key topics include an interpretation of coefficients, various properties of chromatic polynomials, and chromatic equivalence, with an emphasis on understanding the uniqueness of chromatic polynomials across different graphs. The paper aims to contribute to ongoing research in graph theory by analyzing these polynomials and their implications.

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References (65)

  1. 4 Disconnected χ-unique graphs . . . . . . . . . . . . . . . . . 65
  2. 5 One-connected χ-unique graphs . . . . . . . . . . . . . . . . 66
  3. 6 χ-unique graphs with connectivity 2 . . . . . . . . . . . . . 68
  4. 7 A chromatically equivalence class . . . . . . . . . . . . . . . 71
  5. 8 Further results . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.8.1 A K 3 -gluing of graphs . . . . . . . . . . . . . . . . . 73 3.8.2 Polygon-trees and related structures . . . . . . . . . 75 3.8.3 2-connected (n, n + k)-graphs with small k . . . . . 76
  6. Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
  7. 4 Chromaticity of Multi-partite Graphs 83
  8. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
  9. 2 Complete bipartite graphs . . . . . . . . . . . . . . . . . . . 84
  10. 3 Complete tripartite graphs . . . . . . . . . . . . . . . . . . . 86
  11. 4 Complete multi-partite graphs . . . . . . . . . . . . . . . . 90
  12. 5 Complete bipartite graphs with some edges deleted . . . . . 94
  13. 6 Further results . . . . . . . . . . . . . . . . . . . . . . . . . 99
  14. Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
  15. 5 Chromaticity of Subdivisions of Graphs 105
  16. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
  17. 2 Multi-bridge graphs . . . . . . . . . . . . . . . . . . . . . . 106 5.2.1 Chromatic polynomials of multi-bridge graphs . . . 106
  18. 2.2 Generalized polygon-trees . . . . . . . . . . . . . . . 109 5.2.3 Chromaticity of k-bridge graphs with k = 4, 5, 6 . . . 111 5.2.4 Chromaticity of general multi-bridge graphs . . . . . 112
  19. 3 Chromaticity of generalized polygon-trees . . . . . . . . . . 114
  20. 4 K 4 -homeomorphs . . . . . . . . . . . . . . . . . . . . . . . . 118
  21. 5 Chromaticity of uniform subdivisions of graphs . . . . . . . 123
  22. 6 Further results . . . . . . . . . . . . . . . . . . . . . . . . . 127
  23. Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
  24. 6 Graphs in Which any Two Colour Classes Induce a Tree (I) 133
  25. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
  26. 2 The sizes and triangle numbers of graphs in T r . . . . . . . 135
  27. 3 Graphs in T r . . . . . . . . . . . . . . . . . . . . . . . . . . 139
  28. 4 Chordal graphs . . . . . . . . . . . . . . . . . . . . . . . . . 141
  29. 5 q-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Contents XI Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
  30. 7 Graphs in Which any Two Colour Classes Induce a Tree (II) 149
  31. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
  32. 2 The number s 3 (H) . . . . . . . . . . . . . . . . . . . . . . . 151
  33. 3 The family T 3,1 . . . . . . . . . . . . . . . . . . . . . . . . . 153
  34. 4 The structure of graphs in T r,1 (r ≥ 4) . . . . . . . . . . . . 156
  35. 5 Chromatically unique graphs in T r,1 . . . . . . . . . . . . . 161
  36. 6 Further results . . . . . . . . . . . . . . . . . . . . . . . . . 164
  37. Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
  38. 8 Graphs in Which All but One Pair of Colour Classes Induce Trees (I) 169
  39. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
  40. 2 The triangle number and an upper bound . . . . . . . . . . 170
  41. 3 Graphs in F r having maximum triangle numbers . . . . . . 172
  42. 4 A more general result . . . . . . . . . . . . . . . . . . . . . 177
  43. Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
  44. 9 Graphs in Which All but One Pair of Colour Classes Induce Trees (II)
  45. 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
  46. 2 Classification of graphs satisfying (CT) . . . . . . . . . . . 180
  47. 3 Graphs in CT . . . . . . . . . . . . . . . . . . . . . . . . . . 184
  48. 4 Graphs containing exactly one pure cycle . . . . . . . . . . 185
  49. 5 The main results . . . . . . . . . . . . . . . . . . . . . . . . 188
  50. 6 Further results . . . . . . . . . . . . . . . . . . . . . . . . . 190
  51. Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
  52. 10 Chromaticity of Extremal 3-colourable Graphs 195 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.2 3-colourable graphs . . . . . . . . . . . . . . . . . . . . . . . 197
  53. 3 A family of 3-colourable graphs . . . . . . . . . . . . . . . . 200 10.4 Chromaticity of graphs in X k . . . . . . . . . . . . . . . . . 207 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 XII Contents 11 Polynomials Related to Chromatic Polynomials 215 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.2 Basic properties of adjoint polynomials . . . . . . . . . . . . 217 11.3 Reduction formulas for adjoint polynomials . . . . . . . . . 220 11.4 Roots of adjoint polynomials . . . . . . . . . . . . . . . . . 222 11.5 Invariants for adjointly equivalent graphs . . . . . . . . . . 226 11.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 226 11.5.2 The adj-invariant R 1 (G) . . . . . . . . . . . . . . . . 227 11.5.3 The adj-invariant R 2 (G) . . . . . . . . . . . . . . . . 231
  54. 6 Adjointly equivalent graphs . . . . . . . . . . . . . . . . . . 233 11.7 Further results . . . . . . . . . . . . . . . . . . . . . . . . . 241 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 12 Real Roots of Chromatic Polynomials 249 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 12.
  55. Root-free intervals for all chromatic polynomials . . . . . . 250 12.3 Real numbers which are not chromatic roots . . . . . . . . . 256 12.4 Upper root-free intervals . . . . . . . . . . . . . . . . . . . . 256 12.5 Planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 258 12.
  56. Near-triangulations . . . . . . . . . . . . . . . . . . . . . . . 260 12.7 Graphs with hamiltonian paths . . . . . . . . . . . . . . . . 263 12.8 Bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . 264 12.
  57. 9 Graphs containing spanning q-trees . . . . . . . . . . . . . . 266 12.10Largest non-integral chromatic root . . . . . . . . . . . . . . 267 12.11Upper root-free intervals with respect to maximum degrees 269 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 13 Integral Roots of Chromatic Polynomials 273 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 13.2 Chromatic polynomials possessed only by chordal graphs . . 275 13.
  58. Graphs G ∈ I of order ω(G) + 2 . . . . . . . . . . . . . . . 277 13.
  59. Dmitriev's Problem . . . . . . . . . . . . . . . . . . . . . . . 282 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 14 Complex Roots of Chromatic Polynomials 289
  60. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
  61. 2 Location of chromatic roots . . . . . . . . . . . . . . . . . . 290 14.3 Chromatic roots within |z| ≤ 8∆ . . . . . . . . . . . . . . . 293
  62. 4 Subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
  63. 5 Chromatic roots in the whole plane . . . . . . . . . . . . . . 297
  64. 6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 14.
  65. Potts Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 15 Inequalities on Chromatic Polynomials 309 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 15.2 Bounds of chromatic polynomials . . . . . . . . . . . . . . . 310 15.3 Maximum chromatic polynomials . . . . . . . . . . . . . . . 313 15.4 An open problem . . . . . . . . . . . . . . . . . . . . . . . . 318 15.5 Mean colour numbers . . . . . . . . . . . . . . . . . . . . . 321 Exercise 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

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