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5 - Edge-colourings

Published online by Cambridge University Press:  05 May 2015

By
Jessica McDonald
Edited by
Lowell W. Beineke  and
Robin J. Wilson
Jessica McDonald
Affiliation:
Auburn University
Lowell W. Beineke
Affiliation:
Purdue University, Indiana
Robin J. Wilson
Affiliation:
The Open University, Milton Keynes
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Topics in Chromatic Graph Theory , pp. 94 - 113
Publisher: Cambridge University Press
Print publication year: 2015

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References

1. L., Andersen, On edge-colourings of graphs, Math. Scand. 40 (1977), 161–175.Google Scholar
2. L. W., Beineke and R. J., Wilson, On the edge-chromatic number of a graph, Discrete Math. 5 (1973), 15–20.Google Scholar
3. G., Chen, X., Yu and W., Zang, Approximating the chromatic index of multigraphs, J. Combin. Optim. 21 (2011), 219–246.Google Scholar
4. A. G., Chetwynd and A. J. W., Hilton, Regular graphs of high degree are 1-factorizable, Proc. London Math. Soc. (2) 50 (1985), 193–206.Google Scholar
5. A. G., Chetwynd and A. J. W., Hilton, Star multigraphs with three vertices of maximum degree, Math. Proc. Cambridge Philos. Soc. 100 (1986), 303–317.Google Scholar
6. A. G., Chetwynd and A. J. W., Hilton, The edge chromatic class of graphs with maximum degree at least |V| - 3, Graph Theory in Memory of G. A. Dirac (eds. L., Andersen, I., Jakobsen, C., Thomassen, B., Toft and P., Vestergaard), Vol. 1, Ann. Discrete Math. 41, North-Holland (1989), 91–110.Google Scholar
7. A. G., Chetwynd and A. J. W., Hilton, 1-factorizing regular graphs of high degree – an improved bound, Discrete Math. 75 (1989), 103–112.Google Scholar
8. S., Choudum and K., Kayathri, An extension of Vizing's adjacency lemma on edge chromatic critical graphs, Discrete Math. 206 (1999), 97–103.Google Scholar
9. M., Chudnovsky, K., Edwards, K., Kawarabayashi and P., Seymour, Edge-colouring seven-regular planar graphs, to appear.
10. M., Chudnovsky, K., Edwards and P., Seymour, Edge-colouring eight-regular planar graphs, to appear.
11. W. J., Cook, W. H., Cunningham, W. R., Pulleyblank and A., Schrijver, Combinatorial Optimization, Wiley, 1998.Google Scholar
12. Z., Dvořák, K., Kawarabayashi and D., Kral, Packing six T-joins in plane graphs, manuscript.
13. J., Edmonds, Maximum matching and a polyhedron with 0, 1-vertices, J. Research Nat. Bureau of Standards (B) 69 (1965), 125–130.Google Scholar
14. K., Edwards, Optimization and Packings of T-joins and T-cuts, M.Sc. thesis, McGill University, 2011.Google Scholar
15. P., Erdős and R. J., Wilson, On the chromatic index of almost all graphs, J. Combin. Theory (B) 23 (1977), 255–257.Google Scholar
16. L. M., Favrholdt, M., Stiebitz and B., Toft, Graph Edge Colouring: Vizing's Theorem and Goldberg's Conjecture, Preprints 2006, No. 20, IMADA, University of Southern Denmark, 91 pages.
17. S., Fiorini and R. J., Wilson, Edge-colourings of Graphs, Research Notes in Mathematics 16, Pitman, 1977.
18. M. K., Goldberg, On multigraphs with almost-maximal chromatic class [in Russian], Diskret. Analiz 23 (1973), 3–7.Google Scholar
19. M. K., Goldberg, A remark on the chromatic class of a multigraph [in Russian], Vycisl. Mat. i. Vycisl. Tehn. (Kharkow) 5 (1974), 128–130.Google Scholar
20. M. K., Goldberg, Structure of multigraphs with restrictions on the chromatic class [in Russian], Metody Diskret. Analiz 30 (1977), 3–12.Google Scholar
21. M. K., Goldberg, Edge-coloring of multigraphs: recolouring technique, J. Graph Theory 8 (1984), 123–137.Google Scholar
22. M., Grötschel, L., Lovász and A., Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988.Google Scholar
23. S., Grünewald, Chromatic Index Critical Graphs and Multigraphs, Ph.D. thesis, Univer-sitat Bielefeld, 2000.Google Scholar
24. S., Grünewald and E., Steffen, Chromatic-index critical graphs of even order, J. Graph Theory 30 (1999), 27–36.Google Scholar
25. S., Grünewald and E., Steffen, Independent sets and sets and 2-factors in edge-chromatic-critical graphs, J. Graph Theory 45 (2004), 113–118.Google Scholar
26. B., Guenin, Packing T-joins and edge-colouring in planar graphs, manuscript.
27. R. P., Gupta, Studies in the Theories of Graphs, Ph.D. thesis, Tata Institute, Bombay, 1967.Google Scholar
28. P., Haxell and J., McDonald, On characterizing Vizing's edge-colouring bound, J. Graph Theory 69 (2012), 160–168.Google Scholar
29. A. J. W., Hilton and P. D., Johnson, Graphs which are vertex-critical with respect to the edge-chromatic number, Math. Proc. Cambridge Philos. Soc. 102(1987), 211–221.Google Scholar
30. A. J. W., Hilton and R. J., Wilson, Edge-colorings of graphs: a progress report, Graph Theory and its Applications: East and West (Jinan, 1986) (eds. M., Capobianco et al.), Ann. New York Acad. Sci. 576 (1989), 241–249.Google Scholar
31. A. J. W., Hilton and Y., Zhao, One the edge-colouring of graphs whose core has maximum degree two, J. Combin. Math. Combin. Comput. 21 (1996), 97–108.Google Scholar
32. D., Hoffman, Cores of class II graphs, J. Graph Theory 20 (1995), 397–402.Google Scholar
33. D., Hoffman and C., Rodger, Class 1 graphs, J. Combin. Theory (B) 44 (1988), 372–376.Google Scholar
34. D., Hoffman and C., Rodger, The chromatic index of complete multipartite graphs, J. Graph Theory 16 (1992), 159–163.Google Scholar
35. I., Holyer, The NP-completeness of edge-colouring, SIAM J. Comput. 10 (1981), 718–720.Google Scholar
36. I. T., Jakobsen, Some remarks on the chromatic index of a graph, Arch. Math. (Basel) 24 (1973), 440–448.Google Scholar
37. J., Kahn, Asymptotics of the chromatic index for multigraphs, J. Combin. Theory (B) 68 (1996), 223–254.Google Scholar
38. H. A., Kierstead, On the chromatic index of multigraphs without large triangles, J. Combin. Theory (B) 36 (1984), 156–160.Google Scholar
39. D., König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengen-lehre, Math. Ann. 77 (1916), 453–465.Google Scholar
40. A., Kostochka and M., Stiebitz, Edge-colouring: a new fan equation, manuscript, 2006.
41. O., Kurt, On the Edge Coloring of Graphs, Ph.D. thesis, Ohio State University, 2009.Google Scholar
42. R., Luo and Y., Zhao, Finding the exact bound of the maximum degrees of class two graphs embeddable in a surface of characteristic ε є{–1, –2, –3}, J. Combin. Theory (B) 306 (2008), 1788–1790.Google Scholar
43. O., Marcotte, On the chromatic index of multigraphs and a conjecture of Seymour, II, Proc. DIMACS Workshop, Morristown, N. J. (eds. W., Cook and P. D., Seymour), American Math. Soc. (1990), 245–279.Google Scholar
44. O., Marcotte, Optimal edge-colourings for a class of planar multigraphs, Combinatorics 21 (2001), 361–394.Google Scholar
45. J., McDonald, Multigraphs With High Chromatic Index, Ph.D. thesis, University of Waterloo, 2009.Google Scholar
46. J., McDonald, Achieving maximum chromatic index in multigraphs, Discrete Math. 309(8) (2009) 2077–2084.Google Scholar
47. J., McDonald, On multiples of simple graphs and Vizing's theorem, Discrete Math. 310 (2010), 2212–2214.Google Scholar
48. J., McDonald, On a theorem of Goldberg, J. Graph Theory 68(1) (2011), 8–21.Google Scholar
49. J., McDonald, B., Mohar and D., Scheide, Kempe equivalence of edge-colourings in subcubic and subquartic graphs, J. Graph Theory 70(2) (2012), 226–239.Google Scholar
50. B., Mohar, Kempe equivalence of colorings, Graph Theory in Paris: Conf. in Honour of Claude Berge (eds. J. A., Bondy et al.), Birkhäuser (2006), 287–297.Google Scholar
51. T., Nishizeki and K., Kashiwagi, On the 1.1 edge-coloring of multigraphs, SIAM J. Discrete Math. 3(3) (1990), 391–410.Google Scholar
52. T., Nissen and L., Volkmann, Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree, J. Graph Theory 14 (1990), 225–246.Google Scholar
53. O., Ore, The Four-color Problem, Academic Press, 1967.Google Scholar
54. M. W., Padberg and M. R., Rao, Odd minimum cut-sets and b-matchings, Math. Oper. Res. 7(1) (1982), 67–80.Google Scholar
55. L., Perković and B., Reed, Edge coloring regular graphs of high degree, Discrete Math. 165/166 (1997), 567–578.Google Scholar
56. M., Plantholt, A sublinear bound on the chromatic index of multigraphs, Discrete Math. 202 (1999), 201–213.Google Scholar
57. M., Plantholt, Overfull conjecture for graphs with high minimum degree, J. Graph Theory 47 (2004), 73–80.Google Scholar
58. M., Plantholt, A combined logarithmic bound on the chromatic index of multigraphs, J. Graph Theory 73 (2013), 239–259.Google Scholar
59. M., Plantholt and S., Tipnis, The chromatic index of multigraphs of order at most 10, Discrete Math. 177 (1997), 185–193.Google Scholar
60. M., Plantholt and S., Tipnis, All regular multigraphs of even order and high degree are 1-factorable, Electron. J. Combin. 8(1) (2001), R41.Google Scholar
61. N., Robertson, P., Seymour and R., Thomas, Tutte's edge-coloring conjecture, J. Combin. Theory (B) 70 (1997), 166–183.Google Scholar
62. D., Sanders, P., Seymour and R., Thomas, Edge three-coloring cubic doublecross graphs, in preparation.
63. D., Sanders and R., Thomas, Edge three-coloring cubic apex graphs, in preparation.
64. D., Sanders and Y., Zhao, On the size of edge chromatic critical graphs, J. Combin. Theory (B) 86 (2002), 408–412.Google Scholar
65. D., Sanders and Y., Zhao, Coloring edges of graphs embedded in a surface of characteristic zero, J. Combin. Theory (B) 87 (2003), 254–263.Google Scholar
66. D., Scheide, On the 15/14 edge-colouring of multigraphs, Preprints 2007 No. 11, IMADA, University of Southern Denmark, 49 pages.
67. D., Scheide, Edge-colourings of Multigraphs, Ph.D. thesis, TU Ilmenau, 2009.Google Scholar
68. D., Scheide, A polynomial-time edge-colouring algorithm, Preprints 2009 No. 4, IMADA, University of Southern Denmark, 15 pages.
69. D., Scheide, Graph edge-colouring: Tashkinov trees and Goldberg's conjecture, J. Combin. Theory (B) 100 (2010), 68–96.Google Scholar
70. D., Scheide and M., Stiebitz, On Vizing's bound for the chromatic index of a multigraph, Discrete Math. 309 (2009), 4920–4925.Google Scholar
71. A., Schrijver, Theory of Linear and Integer Programming, Wiley, 1986.Google Scholar
72. P., Seymour, Matroids, Hypergraphs and the Max.-Flow Min.-Cut Theorem, D.Phil. Thesis, Oxford University, 1975.Google Scholar
73. P., Seymour, On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math. Soc. 3 (1979), 423–460.Google Scholar
74. P., Seymour, Colouring series-parallel graphs, Combinatorica 10 (1990), 379–392.Google Scholar
75. C. E., Shannon, A theorem on colouring the lines of a network, J. Math. Phys. 28 (1949), 148–151.Google Scholar
76. E., Steffen, A refinement of Vizing's theorem, Discrete Math. 218 (2000), 289–291.Google Scholar
77. M., Stiebitz, D., Scheide, B., Toft and L., Favrholt, Graph Edge-colouring: Vizing's Theorem and Goldberg's Conjecture, Wiley, 2012.Google Scholar
78. V. A., Tashkinov, On an algorithm to color the edges of a multigraph (in Russian), Diskret. Analiz 7 (2000), 72–85.Google Scholar
79. C., Thomassen, Hajiós' conjecture for line graphs, J. Combin. Theory (B) 97 (2007), 156–157.Google Scholar
80. E., Vaughan, An asymptotic version of the multigraph 1-factorization conjecture, J. Graph Theory 72 (2013), 19–29.Google Scholar
81. V. G., Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz 3 (1964), 25–30.Google Scholar
82. V. G., Vizing, Critical graphs with a given chromatic class, Diskret. Analiz 5 (1965), 9–17.Google Scholar
83. V. G., Vizing, The chromatic class of a multigraph, Kibernetika (Kiev) 1 (1965) 29–39 [in Russian]; English transl.: Cybernetics 1 (1965), 32–41.Google Scholar
84. V. G., Vizing, Some unsolved problems in graph theory (in Russian), Uspekhi Mat. Nauk. 23 (1968), 117–134; English transl.: Russian Math. Surveys 23 (1968), 125–141.Google Scholar
85. D. R., Woodall, The average degree of an edge-chromatic critical graph II, J. Graph Theory 56 (2007), 194–218.Google Scholar
86. D. R., Woodall, The independence number of an edge-chromatic critical graph, J. Graph Theory 66 (2011), 98–103.Google Scholar
87. H. P., Yap, Some Topics in Graph Theory, London Math. Soc. Lecture Notes 108, Cambridge Univ. Press, 1986.Google Scholar
88. L., Zhang, Every planar graph with maximum degree 7 is class 1, Graphs Combin. 16 (2000), 467–495.Google Scholar

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