A survey on Hedetniemi's conjecture

Taiwanese Journal of Mathematics

Volumn 2, Issue 1, 1998, Pages 1-24

  Zhu, Xuding ^a  

^a NATIONAL SUN YAT SEN UNIVERSITY   (Taiwan)

Author keywords

Chromatic number; Graph product; Hedetniemi conjecture

Indexed keywords

EID: 0002584352     PISSN: 10275487     EISSN: None     Source Type: Journal    
DOI: 10.11650/twjm/1500406890     Document Type: Review

References (60)

1. J. Bang-Jansen and P. Hell, The effect of two cycles on the complexity of colourings by digraphs, Discrete Appl. Math. 26 (1990), 1-23.
2. J. Bang-Jansen, P. Hell and G. MacGillivray, The complexity of colorings by semicomplete digraphs, SIAM J. Discrete Math. 1 (1988), 281-289.
3. J. Bang-Jensen, P. Hell and G. MacGillivray, On the complexity of colouring by superdigraphs of bipartite graphs, Discrete Math. 109 (1992), 27-44.
4. J. Bang-Jensen, P. Hell and G. MacGillivray, Hereditarily hard colouring problems, submitted to Discrete Math.
5. C. Berge, Graphs and Hypergraphs, North-Holland Publ. Co., 1973.
6. G. Birkhoff, Extended arithmetic, Duke Math. J. 3 (1937), 311-316.
7. G. Birkhoff, Generalized arithmetic, Duke Math. J. 9 (1942), 283-302.
8. B. Bollobas, Uniquely colorable graphs, J. Combin. Theory Ser. B 25 (1978), 54-61.
9. B. Bollobas and N. Sauer, Uniquely colorable graphs with large girth, Canad. J. Math. 28 (1972), 1340-1344.
10. M. Borowiecki, On chromatic number of products of two graphs, Colloq. Math. 25 (1972), 49-52.
11. S. Burr, P. Erdös and L. Lovász, On graphs of Ramsey type, Ars Combin. 1 (1976), 167-190.
12. G. Bloom and S. Burr, On unavoidable digraphs in orientations of graphs, J. Graph Theory 11 (1987), 453-462.
13. D. Duffus, B. Sands and R. Woodrow, On the chromatic number of the products of graphs, J. Graph Theory 9 (1985), 487-495.
14. D. Duffus and N. Sauer, Lattice arising in categorical investigations of Hedetniemi's conjecture, Discrete Math. 152 (1996), 125-139.
15. E. El-Zahar and N. Sauer, The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica 5 (1985), 121-126.
16. D. Greenwell and L. Lovász, Applications of product colouring, Acta Math. Hungar. 25 (1974), 335-340.
17. 1-chromatic graphs can be countable, Combinatorica 5 (1985), 137-139.
18. G. Hajós, Über eine Konstruktion nicht n-färbbarer Graphen, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math. Natur. Rcihe, 10 (1961), 116-117.
19. R. Häggkvist, P. Hell, D. J. Miller, and V. Neumann Lara, On multiplicative graphs and the product conjecture, Combinatorica 8 (1988), 71-81.
20. G. Hahn and C. Tardif, Graph homomorphisms: structure and symmetry, preprint, 1997.
21. C. C. Harner and R. C. Entringer, Arc colorings of digraphs, J. Combin. Theory Ser. B 13 (1972), 219-225.
22. S. T. Hedetniemi, Homomorphisms and graph automata, University of Michigan Technical Report 03105-44-T, 1966.
23. P. Hell, An introduction to the category of graphs, Ann. New York Acad. Sci. 328 (1979), 120-136.
24. P. Hell and D. J. Miller, Graphs with forbidden homomorphic images, Ann. New York Acad. Sci. 319 (1979), 270-280.
25. P. Hell and J. Nešetřil, On the complexity of H-coloring, J. Combin. Theory Ser. B 48 (1990), 92-110.
26. P. Hell and J. Nešetřil, Images of rigid digraphs, European J. Combin. 12 (1991), 33-42.
27. P. Hell and J. Nešetřil, Homomorphisms of graphs and their orientations, Monatshefte für Math. 85 (1978), 39-48.
28. P. Hell and J. Nešetřil, The core of a graph, Discrete Math. 109 (1992), 117-126.
29. P. Hell, J. Nešetřil, and X. Zhu, Duality of graph homomorphisms, Combinatorics, Paul Erdös is Eighty, Volume 2, 271-282. Bolyai Society Mathematical Studies, Budapest, Hungary. 1996.
30. P. Hell, H. Zhou, and X. Zhu, Homomorphisms to oriented cycles, Combinatorica 13 (1993), 421-433.
31. P. Hell, H. Zhou, and X. Zhu, A note on homomorphisms to acyclic local tournaments, J. Graph Theory 20 (1995), 467-471.
32. P. Hell, H. Zhou, and X. Zhu, Multiplicativity of oriented cycles, J. Combin. Theory (B), 60 (1994), 239-253.
33. P. Hell and X. Zhu, Homomorphisms to oriented paths, Discrete Math. 132 (1994), 107-114.
34. P. Hell and X. Zhu, The existence of homomorphisms to oriented cycles, SIAM J. Discrete Math. 8 (1995), 208-222.
35. W. Imrich and S. Klavžar, Graphs Products and Algorithms, book in preparation.
36. S. Klavžar, Coloring of graph products - A survey, Discrete Math. 155 (1996), 135-145.
37. L. Lovász, Operations with structures, Acta. Math. Hungar. 18 (1967), 321-328.
38. D. J. Miller, The categorical product of graphs, Canad. J. Math. 20 (1968), 1511-1521.
39. J. Nešetřil and X. Zhu, Path homomorphisms, Math. Proc. Combridge Philos. Soc. 120 (1996), 207-220.
40. S. Poljak, Coloring digraphs by iterated antichains, Comment. Math. Univ. Carolin. 32 (1992) 209-212.
41. S. Poljak and V. Rödl, On the arc-chromatic number of a digraph, J. Graph Theory, 3 (1981), 339-350.
42. J. Nešetřil and A. Pultr, On classes of relations and graphs determined by subobjects and factorobjects, Discrete Math. 22 (1978), 287-300.
43. G. Sabiddusi, Graph multiplication, Math. Z. 72 (1960), 446-457.
44. N. Sauer and X. Zhu, An approach to Hedetniemi's conjecture, J. Graph Theory, 16 (1992), 423-436.
45. N. Sauer and X. Zhu, Multiplicative posets, Order 8 (1992), 349-358.
46. D. Turzik, A note on chromatic number of direct product of graphs, Comment. Math. Univ. Carolin. 24 (1983), 461-463.
47. E. Welzl, Color families are dense, Theoret. Comput. Sci. 17 (1982), 29-41.
48. E. Welzl, Symmetric graphs and interpretations, J. Combin. Theory Ser. B 37 (1984), 235-244.
49. A. Vince, Star chromatic number, J. Graph Theory 12 (1988), 551-559.
50. H. Zhou, Homomorphism Properties of Graph Products, Ph.D. thesis, Simon Fraser University, 1988.
51. H. Zhou, Multiplicativity, part I - variations, multiplicative graphs and digraphs, J. Graph Theory 15 (1991), 469-488.
52. H. Zhou, Multiplicativity, part II - non-multiplicative digraphs and characterization of oriented paths, J. Graph Theory 15 (1991), 489-509.
53. H. Zhou, On the non-multiplicativity of oriented cycles, SIAM J. Discrete Math. 5 (1992), 207-218.
54. H. Zhou, On the multiplicativity of digraphs, preprint.
55. H. Zhou and X. Zhu, On the multiplicativity of acyclic local tournaments, Combinatorica, to appear.
56. X. Zhu, A polynomial algorithm for homomorphisms to oriented cycles, J. Algorithms 19 (1995), 333-345.
57. X. Zhu, A simple proof of the multiplicativity of directed cycles of prime power length, Discrete Appl. Math. 36 (1992), 313-315.
58. X. Zhu, Multiplicative Structures, Ph.D. thesis, The University of Calgary, 1990.
59. X. Zhu, On the chromatic number of the product of hypergraphs, Ars Combin. 34 (1992), 25-31.
60. X. Zhu, Star chromatic numbers and products of graphs, J. Graph Theory 16 (1992), 557-569.