# On the coloration of perfect graphs

• F. Maffray
Chapter
Part of the CMS Books in Mathematics / Ouvrages de mathématiques de la SMC book series (CMSBM)

## Abstract

We consider only finite graphs, without loops. Given an undirected graph G = (V, E), a k-coloring of the vertices of G is a mapping c: V → {1, 2,..., k} for which every edge xy of G has c(x)c(y). If c(v) = i we say that v has color i. Those sets c-1(T) (i = 1,..., k) that are not empty are called the color classes of the coloring c. Each color class is clearly a stable set (i.e., a subset of vertices with no edge between any two of them), hence we will frequently view a coloring as a partition into stable sets. The graph G is called k-colorable if it admits a k-coloring, and the chromatic number of G, denoted by χ(G), is the smallest integer k such that G is k-colorable. We refer to [9, 16, 29] for general results on graph theory.

## References

1. [1]
H. Aït Haddadène, S. Gravier. On weakly diamond-free Berge graphs. Disc. Math. 159 (1996), 237–240.
2. [2]
H. Aït Haddadène, S. Gravier, F. Maffray. An Algorithm for coloring some perfect graphs. Disc. Math. 183 (1998), 1–16.
3. [3]
H. Aït Haddadène, F. Maffray. Coloring degenerate perfect graphs. Disc. Math. 163 (1997), 211–215.
4. [4]
S.R. Arikati, U.N. Peled. A polynomial algorithm for the parity path problem on perfectly orientable graphs. Disc. App. Math. 65 (1996), 5–20.
5. [5]
S.R. Arikati, C. Pandu Rangan. An efficient algorithm for finding a two-pair, and its applications. Disc. App. Math. 31 (1991), 71–74.
6. [6]
L.W. Beineke. Characterizations of derived graphs. J. Comb. Th. 9 (1970), 129–135.
7. [7]
C. Berge, Les problèmes de coloration en théorie des graphes. Publ. Inst. Stat. Univ. Paris 9 (1960), 123–160.
8. [8]
C. Berge. Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind (Zusammenfassung). Wiss. Z. Martin Luther Univ. Math.-Natur. Reihe, 10 (1961), 114–115.Google Scholar
9. [9]
C. Berge. Graphs. North-Holland, Amsterdam/New York, 1985.
10. [10a]
C. Berge, V. Chvátal (editors). Topics on Perfect Graphs. (1984), North Holland, Amsterdam.
11. [10b]
Ann. Disc. Math. 21 (1984), North Holland, Amsterdam.Google Scholar
12. [11a]
C. Berge, P. Duchet. Strongly perfect graphs. In Topics on Perfect Graphs, C. Berge and V. Chvátal, editors, (1984), North Holland, Amsterdam.Google Scholar
13. [11b]
C. Berge, P. Duchet. Strongly perfect graphs. Ann. Disc. Math. 21 (1984), 57–62.
14. [12]
M.E. Bertschi. La colorabilité unique dans les graphes parfaits, PhD thesis, Math. Institute, University of Lausanne, Switzerland, 1988.Google Scholar
15. [13]
M. E. Bertschi, Perfectly contractile graphs. J. Comb. Th. B 50 (1990), 222–230.
16. [14]
M. E. Bertschi, B.A. Reed. A note on even pairs. Disc. Math. 65 (1987), 317–318.
17. [15]
D. Bienstock, On the complexity of testing for odd holes and odd induced paths. Disc. Math. 90 (1991), 85–92.
18. [16]
B. Bollobás. Modern Graph Theory. Grad. Texts in Math. 184, Springer, 1998.
19. [17]
R.L. Brooks. On colouring the nodes of a network. Proc. Cambridge Phil. Soc. 37 (1941), 194–197.
20. [18]
K. Cameron.Antichain sequences. Order, 2 (1985), 249–255.
21. [19a]
V. Chvátal, Perfectly ordered graphs, In Topics on Perfect Graphs, C. Berge and V. Chvátal, editors, (1984), North Holland, Amsterdam.Google Scholar
22. [19b]
V. Chvátal, Perfectly ordered graphs, Ann. Disc. Math. 21 (1984), 63–68.Google Scholar
23. [20]
V. Chvátal, Star cutsets. J. Comb. Th. B 39 (1985), 189–199.
24. [21]
V. Chvátal, N. Sbihi. Bull-free Berge graphs are perfect. Graphs and Combin. 3 (1987), 127–139.
25. [22]
C.M.H. de Figueiredo, S. Gravier, C. Linhares Sales. On Tucker’s proof of the Strong Perfect Graph Conjecture for K 4-e-free graphs. To appear in Disc. Math. Google Scholar
26. [23]
C.M.H. de Figueiredo, F. Maffray. Optimizing bull-free perfect graphs. Manuscript, Universidade Federal do Rio de Janeiro, Brazil, 1998. To appear in Graphs and Combinatorics. Google Scholar
27. [24]
C.M.H. de Figueiredo, F. Maffray, O. Porto. On the structure of bull-free perfect graphs. Graphs and Combin. 13 (1997), 31–55.
28. [25]
C.M.H. de Figueiredo, F. Maffray, O. Porto. On the structure of bull-free perfect graphs, 2: the weakly triangulated case. RUTCOR Research Report 45–94, Rutgers University, 1994. To appear in Graphs and Combinatorics. Google Scholar
29. [26]
C.M.H. de Figueiredo, J. Meidanis, C. Mello. On edge-colouring indifference graphs. Theor. Comp. Sci. 181 (1997), 91–106.
30. [27]
C.M.H. de Figueiredo, J. Meidanis, C. Mello.Local conditions for edge-coloring. J. Comb. Math, and Comb. Comp. 32 (2000), 79–91.
31. [28]
C.M.H. de Figueiredo, K. Vusšković.A class of beta-perfect graphs. Disc. Math. 216 (2000), 169–193.
32. [29]
R. Diestel. Graph Theory. Grad. Texts in Math. 173, Springer, 1998.Google Scholar
33. [30]
G.A. Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg, 25 (1961), 71–76.
34. [31]
R.D. Dutton, R.C. Brigham. A new graph coloring algorithm. Computer Journal 24 (1981), 85–86.
35. [32]
Th. Emden-Weinert, S. Hougardy, B. Kreuter. Uniquely colourable graphs and the hardness of colouring graphs of large girth. Comb., Prob. & Comp. 7 (1998), 375–386.
36. [33]
P. Erdős. Graph theory and probability. Canad. J. Math. 11 (1959), 34–38.
37. [34]
H. Everett, C.M.H. de Figueiredo, C. Linhares-Sales, F. Maffray, O. Porto, B.A. Reed. Path parity and perfection. Disc. Math. 165/166 (1997), 223–242.Google Scholar
38. [35]
H. Everett, C.M.H. de Figueiredo, C. Linhares-Sales, F. Maffray, O. Porto, B.A. Reed.Even pairs. To appear in Perfect Graphs, J. L. RamírezAlfonsín and B.A. Reed, ed., John Wiley and Sons, 2001.Google Scholar
39. [36]
J. Fonlupt, J.P. Uhry. Transformations which preserve perfectness and h-perfectness of graphs. Ann. Disc. Math. 16 (1982), 83–85.
40. [37]
T. Gallai.Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hungar. 18 (1967), 25–66.
41. [38]
M. Garey, D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979).
42. [39]
G.S. Gasparian. Minimal imperfect graphs: a simple approach. Combinatorica 16 (1996), 209–212.
43. [40]
A. Ghouila-Houri. Caractérisation des graphes non orientés dont on peut orienter les arêtes de manière à obtenir le graphe d’une relation d’ordre. C.R. Acad. Sci. Paris 254 (1962), 1370–1371.
44. [41]
P.C. Gilmore, A.J. Hoffman. A characterization of comparability graphs and of interval graphs. Canadian J. Math. 16 (1964), 539–548.
45. [42]
M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York (1980).
46. [43]
S. Gravier. On Tucker vertices of graphs. Disc. Math. 203 (1999), 121–131.
47. [44a]
M. Grötschel, L. Lovász, A. Schrijver. Polynomial algorithms for perfect graphs. In Topics on Perfect Graphs, C. Berge and V. Chvátal, editors, (1984), North Holland, Amsterdam.Google Scholar
48. [44b]
M. Grötschel, L. Lovász, A. Schrijver. Polynomial algorithms for perfect graphs. Ann. Disc. Math. 21 (1984), 325–356.Google Scholar
49. [45]
R. Hayward, Weakly triangulated graphs. J. Comb. Th. B 39 (1985), 200–208.
50. [46a]
R. Hayward, C.T. Hoàng, F. Maffray. Optimizing weakly triangulated graphs. Graphs and Combin., 5 (1989), 339–349.
51. [46b]
R. Hayward, C.T. Hoàng, F. Maffray. Optimizing weakly triangulated graphs. Graphs and Combin., Erratum in vol. 6 (1990), 33–35.
52. [47]
A. Hertz, A fast algorithm for coloring Meyniel graphs. J. Comb. Th. B 50 (1990), 231–240.
53. [48]
A. Hertz, COSINE, a new graph coloring algorithm. Operations Research Letters 10 (1991), 411–415.
54. [49]
A. Hertz, D. de Werra. Perfectly orderable graphs are quasi-parity graphs: a short proof. Disc. Math. 68 (1988), 111–113.
55. [50]
C.T. Hoàng. Alternating orientation and alternating coloration of perfect graphs. J. Comb. Th. B 42 (1987), 264–273.
56. [51]
C.T. Hoàng. Algorithms for minimum weighted coloring of perfectly ordered, comparability, triangulated and clique-separable graphs. Disc. Appl. Math. 55 (1994), 133–143.
57. [52]
C.T. Hoàng. Perfectly orderable graphs. To appear in Perfect Graphs, J.L. Ramírez-Alfonsín and B.A. Reed, ed., John Wiley and Sons, 2001.Google Scholar
58. [53]
I. Holyer.The NP-completeness of edge-coloring. SIAM J. Computing 10 (1981), 718–720.
59. [54]
S. Hougardy. Perfekte Graphen. PhD thesis, Institut für Ökonometrie und Operations Research, Rheinische Friedrich Wilhelms Universität, Bonn, Germany, 1991.Google Scholar
60. [55]
W. L. Hsu. Decomposition of perfect graphs. J. Comb. Th. B 43 (1987), 70–94.
61. [56a]
W.L. Hsu, G.L. Nemhauser. Algorithms for maximum weighted cliques, minimum weighted clique covers, and minimum colourings of claw-free perfect graphs. In Topics on perfect graphs, C. Berge, and V. Chvátal ed., North-Holland, Amsterdam, 1984.Google Scholar
62. [56b]
W.L. Hsu, G.L. Nemhauser. Algorithms for maximum weighted cliques, minimum weighted clique covers, and minimum colourings of claw-free perfect graphs. Ann. Disc. Maths 21, 1984.Google Scholar
63. [57]
T.R. Jensen, B. Toft. Graph Coloring Problems. Wiley-Interscience Series in Disc. Math, and Optimization, 1995.
64. [58]
D.S. Johnson. Worts case behavior of graph coloring algorithms. Proc. 5 th Southeastern Conf. on Comb., Graph Th. & Comput., Utilitas Mathematica (Winnipeg, 1979), 513–527.Google Scholar
65. [59]
R.M. Karp.Reducibility among combinatorial problems. In R.E. Miller and J.W. Thatcher, editors, Complexity of computer computations, pages 85–104. Plenum Press, New York, 1972.
66. [60]
H. Kierstead, J.H. Schmerl. The chromatic number of graphs which induce neither K 1, 3 nor K 5-e. Disc. Math. 58 (1986) 253–262.
67. [61]
C. Linhares Sales, F. Maffray. Even pairs in claw-free perfect graphs. J. Comb. Th. B 74 (1998), 169–191.
68. [62]
C. Linhares Sales, F. Maffray, B.A. Reed. On planar perfectly contractile graphs. Graphs and Combin. 13 (1997), 167–187.
69. [63]
L. Lovász. On chromatic number of graphs and set-systems. Acta Math. Hung. 19 (1968), 59–67.
70. [64]
L. Lovász. Normal hypergraphs and the perfect graph conjecture. Disc. Math. 2 (1972), 253–267.
71. [65]
L. Lovász. A characterization of perfect graphs. J. Comb. Th. B, 13 (1972), 95–98.
72. [66]
L. Lovász. Three short proofs in Graph Theory. J. Comb. Th. B 19 (1975), 269–271.
73. [67]
L. Lovász. Perfect Graphs. In Selected Topics in Graph Theory 2, L.W. Beineke and R.J. Wilson ed., Academic Press, 1983, 55–87.Google Scholar
74. [68]
L. Lovász, M.D. Plummer. Matching Theory. Annals of Disc. Maths 29, North-Holland, 1986.Google Scholar
75. [69]
C. Lund, M. Yannakakis. On the hardness of approximating minimization problems. J. Assoc. Comp. Mach. 41 (1994), 960–981.
76. [70]
F. Maffray, O. Porto, M. Preissmann.A generalization of simplicial elimination orderings. J. Graph Th., 23 (1996), 203–208.
77. [71]
F. Maffray, M. Preissmann. On the NP-completeness of the k-colorability problem for triangle-free graphs. Disc. Math. 162 (1996), 313–317.
78. [72]
F. Maffray, M. Preissmann. Sequential colorings and perfect graphs. Disc. Appl. Math. 94 (1999), 287–296.
79. [73]
F. Maffray, M. Preissmann.A translation of Tibor Gallai’s article ‘Transitiv orientierbare Graphen’.To appear in Perfect Graphs, J.L. Ramírez-Alfonsín and B.A. Reed, ed., John Wiley and Sons, 2001.Google Scholar
80. [74]
S.E. Markossian, G.S. Gasparian, B.A. Reed.β-perfect graphs. J. Comb. Th. B 67 (1996), 1–11.
81. [75]
D.W. Matula. A min-max theorem with application to graph coloring. SIAM Rev. 10 (1968), 481–482.Google Scholar
82. [76]
D.W. Matüla, L.L. Beck. Smallest last ordering and clustering and graph coloring algorithms. J. Assoc. Comp. Mach. 30 (1983), 417–427.
83. [77]
R.M. McConnell, J.P. Spinrad.Linear-time modular decomposition and efficient transitive orientation of undirected graphs.Proc. 7th Annual ACM-SIAM Symp. Disc. Algorithms. SIAM, Philadelphia, 1997.Google Scholar
84. [78a]
H. Meyniel. The graphs whose odd cycles have at least two chords. In Topics on Perfect Graphs, C. Berge and V. Chvátal, editors, (1984), North-Holland, Amsterdam.Google Scholar
85. [78b]
H. Meyniel. The graphs whose odd cycles have at least two chords. Ann. Disc. Math. 21 (1984), 115–120.
86. [79]
H. Meyniel. A new property of critical imperfect graphs and some consequences. European J. Comb. 8 (1987), 313–316.
87. [80]
M. Middendorf, F. Pfeiffer. On the complexity of recognizing perfectly orderable graphs. Disc. Math. 80 (1990), 327–333.
88. [81]
M. Molloy, B.A. Reed. Colouring graphs whose chromatic number is near their maximum degree. Lecture Notes in Comp. Sci., vol. 1380 (Proc. LATIN’98 Conf.), 216–225, 1998.
89. [82]
J. Mycielski. Sur le coloriage des graphes. Colloq. Math. 3 (1955), 161–162.
90. [83]
J.L. Ramírez-Alfonsín, B.A. Reed (editors). Perfect Graphs. John Wiley and Sons, 2001.
91. [84]
B.A. Reed. Problem session on parity problems (Public communication). DIM ACS Workshop on Perfect Graphs, Princeton University, New Jersey, 1993.Google Scholar
92. [85]
B.A. Reed. A strengthening of Brooks’s theorem. J. Comb. Th. B 76 (1999), 136–149.
93. [86]
F. Roussel, I. Rusu. An O(n 2 ) algorithm to color Meyniel graphs. Manuscript, LIFO, University of Orléans, France, 1998.Google Scholar
94. [87]
J. Spencer. Ten Lectures on the Probabilistic Method. CMBS-NSF Region. Conf. Ser. in Appl. Math., SIAM, Philadelphia, 1994.Google Scholar
95. [88]
J. Spinrad, R. Sritharan. Algorithms for weakly triangulated graphs. Disc. Appl. Math. 59 (1995), 181–191.
96. [89]
M.M. Syslo. Sequential coloring versus Welsh-Powell bound. Disc. Math. 74 (1989), 241–243.
97. [90]
A. Tucker. Coloring perfect (K 4 -e)-free graphs. J. Comb. Th. B 42 (1987), 313–318.
98. [91]
A. Tucker. A reduction procedure for colouring perfect K 4 -free graphs. J. Comb. Th. B 43 (1987), 151–172.
99. [92]
V.G. Vizing. On an estimate of the chromatic class of a p-graph (in Russian). Diskret. Analiz. 3 (1964), 23–30.
100. [93]
D.J.A. Welsh, M.B. Powell. An upper bound on the chromatic number of a graph and its applications to timetabling problems. Computer J. 10 (1967), 85–87.