Homomorphisms to oriented cycles


We discuss the existence of homomorphisms to oriented cycles and give, for a special class of cyclesC, a characterization of those digraphs that admit, a homomorphism toC. Our result can be used to prove the multiplicativity of a certain class of oriented cycles, (and thus complete the characterization of multiplicative oriented cycles), as well as to prove the membership of the corresponding decision problem in the classNPϒcoNP. We also mention a conjecture on the existence of homomorphisms to arbitrary oriented cycles.

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  1. [1]

    J. Bang-Jensen, andP. Hell: On the effect of two cycles on the complexity of colouring,Discrete Applied Math. 26 (1990), 1–23.

    Google Scholar 

  2. [2]

    J. Bang-Jensen, P. Hell, andG. MacGillivray: The complexity of colouring by semicomplete digraphs,SIAM J. on Discrete Math. 1 (1988), 281–298.

    Google Scholar 

  3. [3]

    J. Bang-Jensen, P. Hell, andG. MacGillivray.: On the complexity of colouring by superdigraphs of bipartite graphs.Discrete Math. 109 (1992), 27–44.

    Google Scholar 

  4. [4]

    J. Bang-Jensen, P. Hell, andG. MacGillivray: Hereditarily hard colouring problems, submitted toDiscrete Math.

  5. [5]

    S. Burr, P. Erdős, andL. Lovász: On graphs of Ramsey type,Ars Comb. 1 (1976), 167–190.

    Google Scholar 

  6. [6]

    D. Duffus, B. Sands, andR. Woodrow: On the chromatic number of the product of graphs,J. Graph Theory 9 (1985), 487–495.

    Google Scholar 

  7. [7]

    H. El-Zahar, andN. Sauer: The chromatic number of the product of two 4-chromatic graphs is 4,Combinatorica,5 (1985), 121–126.

    Google Scholar 

  8. [8]

    W. Gutjahr, E. Welzl, andG. Woeginger Polynomial graph colourings,Discrete Applied Math. 35 (1992), 29–46.

    Google Scholar 

  9. [9]

    R. Häggkvist, P. Hell, D. J. Miller, andV. Neumann-Lara: On multiplicative graphs and the product conjecture,Combinatorica 8, (1988), 71–81.

    Google Scholar 

  10. [10]

    R. Häggkvist, andP. Hell: OnA-mote universal graphs,European J. of Combinatorics 14 (1993), 23–27.

    Google Scholar 

  11. [11]

    F. Harary:Graph Theory, Addison Wesley, 1969.

  12. [12]

    S. Hedetniemi: Homomorphisms and graph automata, University of Michigan Technical Report 03105-44-T, 1966.

  13. [13]

    P. Hell: An introduction to the category of graphs.,Annals of the N. Y. Acad. Sc. 328 (1979) 120–136.

    Google Scholar 

  14. [14]

    P. Hell, andJ. Nešetřil: Homomorphisms of graphs and their orientations.,Monatshefte für Math. 85 (1978), 39–48.

    Google Scholar 

  15. [15]

    P. Hell, andJ. Nešetřil: On the complexity ofH-colouring,J. Combin. Theory B 48 (1990), 92–100.

    Google Scholar 

  16. [16]

    P. Hell, H. Zhou, andX. Zhu: Multiplicativity of oriented cycles,Combin. Theory B, in print.

  17. [17]

    P. Hell, andX. Zhu: Homomorphisms to oriented paths,Discrete Math, in print.

  18. [18]

    P. Hell, andX. Zhu: The existence of homomorphisms to unbalanced, cycles, submitted.

  19. [19]

    P. Komárek: Some new good characterizations of directed graphs,Časopis Pěst. Mat. 51 (1984), 348–354.

    Google Scholar 

  20. [20]

    G. MacGillivray: On the complexity of colourings by vertex-transitive and arctransitive digraphs,SIAM J. Discrete Math. 4 (1991), 397–408.

    Google Scholar 

  21. [21]

    H. A. Maurer, J. H. Sudborough, andE. Welzl: On the complexity of the general colouring problem,Inform. and Control 51 (1981), 123–145.

    Google Scholar 

  22. [22]

    D. J. Miller: The categorical product of graphs,Canad. J. Math. 20 (1968), 1511–1521.

    Google Scholar 

  23. [23]

    J. Nešetřil, andA. Pultr: On classes of relations and graphs determined by subobjects and factorobjects,Discrete Math. 22 (1978), 287–300.

    Google Scholar 

  24. [24]

    N. Sauer, andX. Zhu: An approach to Hedetniemi's, conjecture.,J. Graph Theory, to appear.

  25. [25]

    E. Welzl: Symmetric graphs and interpretations,J. Combin. Th. (B) 37 (1984), 235–244.

    Google Scholar 

  26. [26]

    H. Zhou: Homomorphism properties of graph products, Ph.D. thesis, Simon Fraser University, 1988.

  27. [27]

    X. Zhu: Multiplicative structures, Ph.D. thesis, The University of Calgary, 1990.

  28. [28]

    X. Zhu: A polynomial algorithm for homomorphisms to oriented cycles, manuscript, 1991.

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Hell, P., Zhou, H. & Zhu, X. Homomorphisms to oriented cycles. Combinatorica 13, 421–433 (1993). https://doi.org/10.1007/BF01303514

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AMS subject classification code (1991)

  • 05 C 20
  • 05 C 99