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Combinatorica

, Volume 8, Issue 1, pp 63–74 | Cite as

On multiplicative graphs and the product conjecture

  • R. Häggkvist
  • P. Hell
  • D. J. Miller
  • V. Neumann Lara
Article

Abstract

We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Nešetřil and Pultr These results allow us (in conjunction with earlier results of Nešetřil and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].

AMS subject classification (1980)

05 C 15 05 C 20 05 C 38 

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References

  1. [1]
    S. Burr, P. Erdős andL. Lovász, On graphs of Ramsey type,ARS Comb.,1 (1976), 167–190.Google Scholar
  2. [2]
    G. Bloom andS. Burr, On unavoidable digraphs in orientations of graphs,J. Graph Theory 11 (1987), 453–462.Google Scholar
  3. [3]
    D.Duffus, B.Sands and R.Woodrow, On the chromatic number of the products of graphs,J. Graph Theory, to appear.Google Scholar
  4. [4]
    M. El-Zahar andN. Sauer, The chromatic number of the product of two four-chromatic graphs is four,Combinatorica,5 (1985), 121–126.Google Scholar
  5. [5]
    A. M. H.Gerards, Homomorphisms of graphs into odd cycles,preprint 1986.Google Scholar
  6. [6]
    P. J.Giblin,Graphs, Surfaces and Homology, Chapman and Hall, 1977.Google Scholar
  7. [7]
    R.Häggkvist, P.Hell, D. J.Miller and V.Neumann-Lara, On multiplicative graphs and the product conjecture,Report No. 11, 1985,Matematiska Institutionen, Stockholms Universitet.Google Scholar
  8. [8]
    A. Hajnal, The chromatic number of the product of two aleph-one chromatic graphs can be countable,Combinatorica,5 (1985), 137–139.Google Scholar
  9. [9]
    F.Harary,Graph Theory, Addison-Wesley, 1969.Google Scholar
  10. [10]
    S.Hedetniemi, Homomorphisms of graphs and automata,University of Michigan Technical Report 03105-44-T, 1966.Google Scholar
  11. [11]
    P. Hell,Retracts in graphs, Springer-Verlag Lecture Notes in Mathematics406 (1974), 291–301.Google Scholar
  12. [12]
    P. Hell, Absolute retracts and the four color conjecture,J. Combin. Theory (B),17(1984), 5–10.Google Scholar
  13. [13]
    P. Hell, On some strongly rigid families of graphs and the full embeddings they induce,Alg. Universalis. 4 (1974), 108–126.Google Scholar
  14. [14]
    P. Hell andJ. Nešetřil, Graphs andk-societies,Canad. Math. Bull. 13 (1970), 375–381Google Scholar
  15. [15]
    P. Hell andJ. Nešetřil, Cohomomorphisms of graphs and hypergraphs,Math., Nachr. 87 (1979), 53–61.Google Scholar
  16. [16]
    D. J. Miller, The categorical product of graphs,Canada. J. Math.,20 (1968), 1511–1521.Google Scholar
  17. [17]
    J. Nešetřil andA. Pultr, On classes of relations and graphs determined by subobjects and factorobjects.Discrete Math. 22 (1979), 187–300.Google Scholar
  18. [18]
    R. Nowakowski andI. Rival, Fixed-edge theorem for graphs with loops.J. Graph Theory. 3 (1981), 339–350.Google Scholar
  19. [19]
    S. Poljak andV. Rödl, On the arc-chromatic number of a digraph,J. Combin. Th. (B),31 (1981), 190–198.Google Scholar
  20. [20]
    E. Welzl, Symmetric graphs and interpretations,J. Combin. Th. (B),37 (1984), 235–744.Google Scholar

Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • R. Häggkvist
    • 1
  • P. Hell
    • 3
  • D. J. Miller
    • 2
  • V. Neumann Lara
    • 4
  1. 1.University of StockholmStockholmSweden
  2. 2.University of VictoriaVictoriaUSA
  3. 3.Simon Fraser UniversityBurnabyUSA
  4. 4.University of MexicoMexico CityMexico

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