Advertisement

Graph Symmetry pp 107-166 | Cite as

Graph homomorphisms: structure and symmetry

  • Geňa Hahn
  • Claude Tardif
Part of the NATO ASI Series book series (ASIC, volume 497)

Abstract

This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex-transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a discussion of various graph products.

Keywords

Chromatic Number Cayley Graph Isometric Embedding Petersen Graph Lexicographic Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. L. Abbott, B. Zhou, The star chromatic number of a graph, J. Graph Theory 17 (1993), 349–360.MathSciNetzbMATHGoogle Scholar
  2. [2]
    M. E. Adams, J. Nesetiil, J. Sichler, Quotients of rigid graphs, J. Combin. Theory Ser. B 30 (1981), 351–359.MathSciNetzbMATHGoogle Scholar
  3. [3]
    M. O. Albertson, D. M. Berman, The chromatic difference sequence of a graph, J. Combin. Theory Ser. B 29 (1980), 1–12.MathSciNetGoogle Scholar
  4. [4]
    M. O. Albertson, V. Booth, Homomorphisms of symmetric graphs, Congr. Numer. 53 (1986), 79–86.MathSciNetGoogle Scholar
  5. [5]
    M. O. Albertson, K. L. Collins, Homomorphisms of 3-chromatic graphs, Discrete Math. 54 (1985), 127–132.MathSciNetzbMATHGoogle Scholar
  6. [6]
    M. O. Albertson, P. A. Catlin, L. Gibbob, Homomorphisms of 3-chromatic graphs II, Congr. Numer. 47 (1985), 19–28.MathSciNetGoogle Scholar
  7. [7]
    L. Babai, C. Godsil, On the automorphism group of almost all Cayley graphs, European J. Comb. 3 (1982), 9–15.MathSciNetzbMATHGoogle Scholar
  8. [8]
    L. Babai, J. Nes“etfil, High chromatic rigid graphs I, in: Combinatorics (A. Hajnal, V. T. Sós, eds.), Colloq. Math. Soc. Janos Bolyai 18, North-Holland, Amsterdam, 1978, 53–60.Google Scholar
  9. [9]
    L. Babai, J. Nesetfil, High chromatic rigid graphs II, in: Algebraic and Geometric Combinatorics (E. Mendelsohn, ed.), Ann. Discrete Math. 15 (1982), 55–61.Google Scholar
  10. [10]
    H. J. Bandelt, Retracts of hypercubes, J. Graph Theory 8 (1984), 501–510.MathSciNetzbMATHGoogle Scholar
  11. [11]
    H. J. Bandelt, A. Dählmann, H. Schütte, Absolute retracts of bipartite graphs, Discrete Appl. Math. 16 (1987), 191–215.MathSciNetzbMATHGoogle Scholar
  12. [12]
    J. Bang-Jensen, P. Hell, and G. MacGillivray, The complexity of colouring by semi-complete digraphs, SIAM J. Discrete Math. 1 (1988), 281–298.MathSciNetzbMATHGoogle Scholar
  13. [13]
    J. Bang-Jensen, P. Hell, and G. MacGillivray, Hereditarily hard H-colouring problems, Discrete Math. 138 (1995), 75–92.MathSciNetzbMATHGoogle Scholar
  14. [14]
    I. BârAny, A short proof of Kneser’s conjecture, J. Combin. Theory Ser. A 25 (1978), 325–326.MathSciNetzbMATHGoogle Scholar
  15. [15]
    B. Bauslaugh, Core-like properties of infinite graphs and structures, Discrete Math. 138 (1995), 101–111.MathSciNetzbMATHGoogle Scholar
  16. [16]
    B. Bauslaugh, Homomorphisms in infinite structures, Ph. D. Thesis, Simon Fraser University, 1994.Google Scholar
  17. [17]
    B. Bauslaugh, The complexity of infinite H-colouring, J. Combin. Theory Ser. B 61 (1994), 141–154.MathSciNetzbMATHGoogle Scholar
  18. [18]
    J. A. Bondy, P. Hell, A note on the star chromatic number, J. Graph Theory 14 (1990), 479–482.MathSciNetzbMATHGoogle Scholar
  19. [19]
    J. A. Bondy, U. S. R. Murty, Graph Theory With Applications, Macmillan, London, 1976.zbMATHGoogle Scholar
  20. [20]
    R. Brewster, G. MacGillivray, Homomorphically full graphs, Discrete Appl. Math. 66 (1996), 23–31.MathSciNetzbMATHGoogle Scholar
  21. [21]
    N. de Bruijn, P. Erdfis, A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 371–373 ( Indag. Math. 13 ).Google Scholar
  22. [22]
    F. Buckley, L. Superville, The a-jointed number and graph homomorphism problems, in: The Theory and Applications of Graphs ( G. Chartrand et al., eds.), Wiley, New York, 1981, 149–158.Google Scholar
  23. [23]
    S. A. Burr, P. Erdös, L Lovâsz, On graphs of Ramsey type, Ars Combin. 1 (1976), 167–190.zbMATHGoogle Scholar
  24. [24]
    P. Catlin, Graph homomorphisms onto the five-cycle, J. Combin. Theory Ser. B 45 (1988), 199–211.MathSciNetzbMATHGoogle Scholar
  25. [25]
    P. Catlin, Homomorphisms as generalizations of graph colouring, Congr. Numer. 50 (1985), 179–186.MathSciNetGoogle Scholar
  26. [26]
    G. Chang, L. Huang, X. Zhu, The star-chromatic number of Mycielski’s graphs, preprint, National Sun Yat-sen University, Taiwan, 1996.Google Scholar
  27. [27]
    F. R. K. Chung, Z. Füredi, R. L. Graham, P. Seymour, On induced subgraphs of the cube, J. Combin. Theory Ser. A 49 (1988), 180–187.MathSciNetzbMATHGoogle Scholar
  28. [28]
    F. R. K. Chung, R. L. Graham, M. E. Saks, A dynamic location problem for graphs, Combinatorica 9 (1989), 111–131.MathSciNetzbMATHGoogle Scholar
  29. [29]
    V. Chvâtal, P. Hell, L. Kucera, J. Nes’etfil, Every finite graph is a full subgraph of a rigid graph, J. Combin. Theory 11 (1971), 284–286.zbMATHGoogle Scholar
  30. [30]
    C. R. Cook, A. B. Evans, Graph folding, Congr. Numer. 23 (1979), 305–314.MathSciNetGoogle Scholar
  31. [31]
    D. G. Corneil, A. Wagner, On the complexity of the embedding problem for hypercube related graphs, Discrete Appl. Math. 43 (1993), 75–95.MathSciNetzbMATHGoogle Scholar
  32. [32]
    W. Deuber, X. Zhu, Chromatic numbers of distance graphs, to appear in Discrete Math.Google Scholar
  33. [33]
    W. Deuber, X. Zhu, Relaxed coloring of a graph, to appear in Graphs and Combin.Google Scholar
  34. [34]
    W. Deuber, X. Zhu, Circular coloring of weighted graphs, J. Graph Theory 23 (1996), 365–376.MathSciNetzbMATHGoogle Scholar
  35. [35]
    G. A. Dirac, Homomorphism theorems for graphs, Math. Ann. 153 (1964), 69–80.MathSciNetGoogle Scholar
  36. [36]
    D. 2. Djokovic, Distance-preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973), 263–267.Google Scholar
  37. [37]
    D. Duffus, I. Rival, Graphs orientable as distributive lattices, Proc. Amer. Math. Soc. 88 (1983), 197–200.MathSciNetzbMATHGoogle Scholar
  38. [38]
    D. Duffus, B. Sands, R. E. Woodrow, On the chromatic number of the product of graphs, J. Graph Theory 9 (1985), 487–495.MathSciNetzbMATHGoogle Scholar
  39. [39]
    D. Duffus, N. Sauer, Lattices arising in categorial investigations of Hedetniemi’s conjecture, Discrete Math. 152 (1996), 125–139.MathSciNetzbMATHGoogle Scholar
  40. [40]
    M. El-Zahar, N. Sauer, The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica 5 (1985), 121–126.MathSciNetzbMATHGoogle Scholar
  41. [41]
    P. Erdös, Graph theory and probability, Canad. J. Math. 11 (1959), 34–38.MathSciNetzbMATHGoogle Scholar
  42. [42]
    P. Erdös, C. Ko, R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. 12 (1961), 313–320Google Scholar
  43. [43]
    P. Erdös, J. Spencer, Probabilistic Methods in Combinatorics, Academic Press, New York, 1974.zbMATHGoogle Scholar
  44. [44]
    O. Favaron, personal communication.Google Scholar
  45. [45]
    G. Gao, G. Hahn, Minimal graphs that fold onto Kn, Discrete Math. 142 (1995), 277–280.MathSciNetzbMATHGoogle Scholar
  46. [46]
    G. Gao, G. Hahn, H. Zhou, Star chromatic number of flower graphs, preprint, Université de Montréal, 1992.Google Scholar
  47. [47]
    G. Gao, E. Mendelsohn, H. Zhou, Computing star chromatic number from related graph invariants, J. Combin. Math. Combin. Comput. 16 (1994), 87–95.MathSciNetzbMATHGoogle Scholar
  48. [48]
    G. Gao, X. Zhu, Star-extremal graphs and lexicographic product. Discrete Math. 152 (1996), 147–156.MathSciNetzbMATHGoogle Scholar
  49. [49]
    A. H. M. Gerards, Homomorphisms of graphs into odd cycles, J. Graph Theory 12 (1988), 73–83.MathSciNetzbMATHGoogle Scholar
  50. [50]
    C. D. Godsil, Problems in algebraic combinatorics, Electron. J. Combin. 2 (1995), Feature 1, approx. 20 pp. (electronic).Google Scholar
  51. [51]
    W. H. Gottschalk, Choice functions and Tychonoff’s theorem, Proc. Amer. Math. Soc. 2 (1951), 172.MathSciNetzbMATHGoogle Scholar
  52. [52]
    R. L. Graham, P. M. Winkler, On isometric embeddings of graphs, Trans. Amer. Math. Soc. 288 (1985), 527–536.MathSciNetzbMATHGoogle Scholar
  53. [53]
    D. Guan, X. Zhu, A coloring problem for weighted graphs, to appear in Inform. Process. Lett.Google Scholar
  54. [54]
    R. Häggkvist, P. Hell, D. J. Miller, V. Neumann Lara, On multiplicative graphs and the product conjecture, Combinatorica 8 (1988), 63–74.MathSciNetzbMATHGoogle Scholar
  55. [55]
    G. Hahn, P. Hell, S. Poljak, On the ultimate independence ratio of a graph, European J. Combin. 16 (1995), 253–261.MathSciNetzbMATHGoogle Scholar
  56. [56]
    G. Hahn, G. MacGillivray, Graph homomorphisms II: Computational aspects and infinite graphs, preprint, Université de Montréal, 1997.Google Scholar
  57. [57]
    G. Hahn, J. Sirân, A note on intersecting cliques, J. Combin. Math. Combin. Comput. 18 (1995), 57–63.MathSciNetzbMATHGoogle Scholar
  58. [58]
    A. Hajnal, The chromatic number of the product of two N1-chromatic graphs can be countable, Combinatorica 5 (1985), 137–139.MathSciNetzbMATHGoogle Scholar
  59. [59]
    F. Harary, S. Hedetniemi, Achromatic number of a graph, J. Combin. Theory 8 (1970), 154–161.MathSciNetzbMATHGoogle Scholar
  60. [60]
    F. Harary, S. Hedetniemi, G. Prins, An interpolation theorem for graphical homomorphisms, Portugal. Math. 26 (1967), 453–462.MathSciNetzbMATHGoogle Scholar
  61. [61]
    F. Harary. D. Hsu, Z. Miller, The bichromaticity of a tree, in: Theory and Applications of Graphs (Y. Alavi, D. R. Lick, eds.), Lecture Notes in Math. 642, Springer-Verlag, Berlin, 1978, 236–246.Google Scholar
  62. [62]
    S. Hedetniemi, Homomorphisms of graphs and automata, University of Michigan Technical Report 03105–44-T, 1966.Google Scholar
  63. [63]
    Z. Hedrlín, P. Hell, C. S. Ko, Homomorphism interpolation and approximation, in: Algebraic and Geometric Combinatorics (E. Mendelsohn, ed.), Ann. Discrete Math. 15 (1982), 213–227.Google Scholar
  64. [64]
    P. Hell, Rigid undirected graphs with given number of vertices, Comment. Math. Univ. Carolinae 9 (1968), 51–69.MathSciNetzbMATHGoogle Scholar
  65. [65]
    P. Hell, Rétractions de graphes, Ph. D. Thesis, Université de Montréal, 1972.Google Scholar
  66. [66]
    P. Hell, Absolute retracts in graphs, in: Graphs and Combinatorics (R. A. Bari, F. Harary, eds.), Lecture Notes in Math. 406 (1974), 291–301.Google Scholar
  67. [67]
    P. Hell, An introduction to the category of graphs, Ann. N. Y. Acad. Sciences 328 (1979), 120–136.MathSciNetGoogle Scholar
  68. [68]
    P. Hell, D. Miller, Graphs with given achromatic number, Discrete Math. 16 (1976), 195–207.MathSciNetzbMATHGoogle Scholar
  69. [69]
    P. Hell, D. Miller, On forbidden quotients and the achromatic number, in Congr. Nu-mer. 15 (1976), 283–292.MathSciNetGoogle Scholar
  70. [70]
    P. Hell, D. Miller, Achromatic number and graph operations, Discrete Math. 108 (1992), 297–305.MathSciNetzbMATHGoogle Scholar
  71. [71]
    P. Hell, D. Miller, Graphs with forbidden homomorphic images, Ann. N. Y. Acad. Sci. 319 (1979), 270–280.MathSciNetGoogle Scholar
  72. [72]
    P. Hell, J. Nes’etril, On the complexity of H-colourings, J. Combin. Theory Ser. B 48 (1990), 92–110.MathSciNetzbMATHGoogle Scholar
  73. [73]
    P. Hell, J. Nesetfil, The core of a graph, Discrete Math. 109 (1992), 117–126.MathSciNetzbMATHGoogle Scholar
  74. [74]
    P. Hell, J. Nes“etfil, X. Zhu, Duality of graph homomorphisms, in: Combinatorics, Paul Erdós is Eighty, vol. 2, Bolyai Society Mathematical Studies, Budapest, 1996, 271–282.Google Scholar
  75. [75]
    P. Hell, X. Yu, H. Zhou, Independence ratios of graph powers, Discrete Math. 127 (1994), 213–220.MathSciNetzbMATHGoogle Scholar
  76. [76]
    P. Hell, H. Zhou, X. Zhu, Homomorphisms to oriented cycles, Combinatorica 13 (1993), 421–433.MathSciNetzbMATHGoogle Scholar
  77. [77]
    P. Hell, H. Zhou, X. Zhu, Multiplicative oriented cycles, J. Combin. Theory Ser. B 60 (1994), 239–253.Google Scholar
  78. [78]
    P. Hell, X. Zhu, Homomorphisms to oriented paths, Discrete Math. 132 (1994) 107114.Google Scholar
  79. [79]
    P. Hell, X. Zhu, The existence of homomorphisms to oriented cycles, SIAM J. Discrete Math. 8 (1995), 208–222.MathSciNetzbMATHGoogle Scholar
  80. [80]
    P. Hell, H. Zhou, X. Zhu, A note on homomorphisms to acyclic local tournaments, J. Graph Theory 20 (1995), 467–471.MathSciNetzbMATHGoogle Scholar
  81. [81]
    A. J. W. Hilton, E. C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 2 18 (1967), 369–384.Google Scholar
  82. [82]
    F. Hughes, G. MacGillivray, The achromatic number of graphs: A survey and some new results, Bull. Inst. Combin. Appl. 19 (1997), 27–56.MathSciNetzbMATHGoogle Scholar
  83. [83]
    W. Imrich, J. Zerovnik, Factoring Cartesian-product graphs, J. Graph Theory 18 (1994), 557–567.MathSciNetzbMATHGoogle Scholar
  84. [84]
    W. Imrich, S. Klavzar, Retracts of strong product of graphs, Discrete Math. 109 (1992), 147–154.MathSciNetzbMATHGoogle Scholar
  85. [85]
    S. Klavzar, U. Milutinovié, Strong products of Kneser graphs, Discrete Math. 133 (1994), 297–300.MathSciNetzbMATHGoogle Scholar
  86. [86]
    A. Kostochka, E. Sopena, X. Zhu, Acyclic and oriented chromatic numbers of graphs, to appear in J. Graph Theory.Google Scholar
  87. [87]
    P. Kfivka, On homomorphism perfect graphs, Comment. Math. Univ. Carolinae 12 (1971), 619–626.Google Scholar
  88. [88]
    H. -J. Lai, Unique graph homomorphisms onto odd cycles, Utilitas Math. 31 (1987), 199–208.MathSciNetzbMATHGoogle Scholar
  89. [89]
    H. -J. Lai, Unique graph homomorphisms onto odd cycles II, J. Combin. Theory Ser. B 46 (1989), 363–376.MathSciNetzbMATHGoogle Scholar
  90. [90]
    B. Larose, F. Laviolette, C. Tardif, Normal Cayley graphs and homomorphisms of Cartesian powers of graphs, preprint, Université de Montréal, 1994,submitted to European J. Combin.Google Scholar
  91. [91]
    L. Lovdsz, Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory Ser. A 25 (1978), 319–324.MathSciNetGoogle Scholar
  92. [92]
    H. A. Maurer, A. Salomaa, E. Welzl, On the complexity of the general colouring problem, Inform. and Control 51 (1981), 128–145.MathSciNetzbMATHGoogle Scholar
  93. [93]
    H. A. Maurer, A. Salomaa, D. Wood, Colorings and interpretations: a connection between graphs and grammar forms, Discrete Appl. Math. 3 (1981), 119–135.MathSciNetzbMATHGoogle Scholar
  94. [94]
    J. Nes“ettil, Homomorphisms of derivative graphs, Discrete Math. 1 (1971), 257–268.MathSciNetzbMATHGoogle Scholar
  95. [95]
    J. Nesetfil, X. Zhu, Path homomorphisms, Math. Proc. Cambridge Philos. Soc. 120 (1996), 207–220.MathSciNetGoogle Scholar
  96. [96]
    J. Nes’etfil, X. Zhu, On bounded treewidth duality of graphs, J. Graph Theory 23 (1996), 151–162.MathSciNetGoogle Scholar
  97. [97]
    R. Nowakowski, I. Rival, On a class of isometric subgraphs of a graph, Combinatorica 2 (1982), 79–90.MathSciNetzbMATHGoogle Scholar
  98. [98]
    R. Nowakowski, I, Rival, The smallest graph variety containing all paths, Discrete Math. 43 (1983), 235–239.MathSciNetzbMATHGoogle Scholar
  99. [99]
    M. Perles, personal communication via P. Hell.Google Scholar
  100. [100]
    S. Poljak, Coloring digraphs by iterated antichains, Comment. Math. Univ. Carolinae 32 (1992), 209–212.Google Scholar
  101. [101]
    S. Poljak, V. Rödl, On the arc-chromatic number of a digraph, J. Combin. Theory Ser. B 31 (1981), 190–198.MathSciNetzbMATHGoogle Scholar
  102. [102]
    A. Pultr, V. Trnkovâ, Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, North-Holland, Amsterdam, 1980.Google Scholar
  103. [103]
    A. Quilliot, Un problème de point fixe sur les graphes, Ann. Rev. Inst. Mat. Mexico, 1981.Google Scholar
  104. [104]
    A. Quilliot, On the Helly property working as a compactness criterion on graphs, J. Combin. Theory Ser. A 40 (1985), 186–193.MathSciNetzbMATHGoogle Scholar
  105. [105]
    I. Rival, Maximal sublattices of finite distributive lattices, Proc. Amer. Math. Soc: 37 (1973), 417–420.MathSciNetzbMATHGoogle Scholar
  106. [106]
    J. Rotman, The Theory of Groups, Allyn and Bacon, Boston, 1973.zbMATHGoogle Scholar
  107. [107]
    G. Sabidussi, Graphs with given group and given graph-theoretical properties, Canad. J. Math. 9 (1957), 515–525.MathSciNetzbMATHGoogle Scholar
  108. [108]
    G. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426–438.Google Scholar
  109. [109]
    G. Sabidussi, Graph derivatives, Math. Z. 76 (1961), 385–401.MathSciNetzbMATHGoogle Scholar
  110. [110]
    N. Sauer, X. Zhu, An approach to Hedetniemi’s conjecture, J. Graph Theory 16 (1992), 423–436.MathSciNetzbMATHGoogle Scholar
  111. [111]
    N. Sauer, X. Zhu, Multiplicative posets, Order 8 (1992), 349–358.Google Scholar
  112. [112]
    H. Shapiro, The embedding of graphs in cubes and the design of sequential relay circuits, Bell Telephone Laboratories Memorandum, 1953.Google Scholar
  113. [113]
    S. Stahl, n-tuple colorings and associated graphs, J. Combin. Theory Ser. B 20 (1976), 185–203.Google Scholar
  114. [114]
    S. Stahl, The multichromatic numbers of some Kneser graphs, preprint, University of Kansas, 1996.Google Scholar
  115. [115]
    E. Steffen, X. Zhu, Star chromatic numbers of graphs, Combinatorica 16 (1996), 439448.Google Scholar
  116. [116]
    L. Szamkolowicz, Remarks on the Cartesian product of two graphs, Colloq. Math. 9 (1962), 43–47.MathSciNetzbMATHGoogle Scholar
  117. [117]
    C. Tardif, A fixed box theorem for the cartesian product of graphs and metric spaces, to appear in Discrete Math.Google Scholar
  118. [118]
    C. Tardif, Fractional multiples of graphs and the density of vertex-transitive graphs, preprint, Université de Montréal, 1996.Google Scholar
  119. [119]
    C. Tardif, Graph products and the chromatic difference sequence of vertex transitive graphs, preprint, Université de Montréal, 1996.Google Scholar
  120. [120]
    C. Tardif, Homomorphismes du graphe de Petersen et inégalités combinatoires, preprint, Université de Montréal, 1996.Google Scholar
  121. [121]
    B. Toft, Graph Colouring Problems, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1995.Google Scholar
  122. [122]
    J. Turner, Point-symmetric graphs with a prime number of points, J. Combin. Theory Ser. B 3 (1967), 136–145.zbMATHGoogle Scholar
  123. [123]
    D. Turzik, A note on chromatic number of direct product of graphs, Comment. Math. Univ. Carolinae 24 (1983), 461–463.MathSciNetzbMATHGoogle Scholar
  124. [124]
    K. Vesztergombi, Chromatic number of strong products of graphs, in: Algebraic Methods in Graph Theory, vol. 2 (L. Lovâsz, V. T. Sós, eds.), Colloq. Math. Soc. Janos Bolyai 25, North-Holland, Amsterdam, 1981, 819–825.Google Scholar
  125. [125]
    A. Vince, Star chromatic number, J. Graph Theory 12 (1988), 551–559.MathSciNetzbMATHGoogle Scholar
  126. [126]
    J. Walker, From graphs to ortholattices and equivariant maps, J. Combin. Theory Ser. B 35 (1983), 171–192.MathSciNetzbMATHGoogle Scholar
  127. [127]
    E. Welzl, Color-families are dense, J. Theoret. Comput. Sci. 17 (1982), 29–41.MathSciNetzbMATHGoogle Scholar
  128. [128]
    E. Welzl, Symmetric graphs and interpretations, J. Combin. Theory Ser. B 37 (1984), 235–244.MathSciNetzbMATHGoogle Scholar
  129. [129]
    E. Wilkeit, The retracts of Hamming graphs, Discrete Math. 102 (1992), 197–218.MathSciNetzbMATHGoogle Scholar
  130. [130]
    B. Zelinka, Homomorphisms of finite bipartite graphs onto complete bipartite graphs, Math. Slovaca 33 (1983), 545–547.Google Scholar
  131. [131]
    B. Zelinka, Homomorphisms of infinite bipartite graphs onto complete bipartite graphs, Czechoslovak Math. J. 32 (1982), 361–366.MathSciNetzbMATHGoogle Scholar
  132. [132]
    H. Zhou, Chromatic difference sequences and homomorphisms, Discrete Math. 113 (1993), 285–292.MathSciNetzbMATHGoogle Scholar
  133. [133]
    H. Zhou, The chromatic difference sequence of the cartesian product of graphs, Discrete Math. 90 (1991), 297–311.MathSciNetzbMATHGoogle Scholar
  134. [134]
    H. Zhou, The chromatic difference sequence of the cartesian product of graphs: Part II, Discrete Appl. Math. 41 (1993), 263–267.MathSciNetzbMATHGoogle Scholar
  135. [135]
    H. Zhou, Homomorphism properties of graphs, Ph. D. Thesis, Simon Fraser University, 1988.Google Scholar
  136. [136]
    H. Zhou, X. Zhu, On the multiplicativity of acyclic local tournaments, to appear in Combinatorica.Google Scholar
  137. [137]
    X. Zhu, On the bounds for the ultimate independence ratio of graphs, Discrete Math. 156 (1996), 207–220.Google Scholar
  138. [138]
    X. Zhu, Star chromatic numbers and products of graphs, J. Graph Theory 16 (1992), 557–569.MathSciNetzbMATHGoogle Scholar
  139. [139]
    X. Zhu, On the chromatic number of the products of hypergraphs Ars Combin. 34 (1992), 25–31.MathSciNetzbMATHGoogle Scholar
  140. [140]
    X. Zhu, A simple proof of the multiplicativity of directed cycles of prime power length, Discrete Appi. Math. 36 (1992), 313–315.zbMATHGoogle Scholar
  141. [141]
    X. Zhu, A note on graph reconstruction, to appear in Ars Combinatoria.Google Scholar
  142. [142]
    X. Zhu, Circular chromatic number a survey, preprint, National Sun Yat-sen University, Taiwan, 1997.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Geňa Hahn
    • 1
  • Claude Tardif
    • 2
  1. 1.Département d’informatique et de recherche opérationnelleMontréalCanada
  2. 2.Département de mathématiques et de statistiqueUniversité de MontréalMontréalCanada

Personalised recommendations