On multiplicative graphs and the product conjecture

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].

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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.

  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.

  6. [6]

    P. J.Giblin,Graphs, Surfaces and Homology, Chapman and Hall, 1977.

  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.

  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.

  10. [10]

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

  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), 375–381

    Google 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 

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Parts of this paper were written when the second author visited the University of Stockholm; other parts were written when the last author visited Simon Fraser University. The hospitality of both Universities is gratefully acknowledged.

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Häggkvist, R., Hell, P., Miller, D.J. et al. On multiplicative graphs and the product conjecture. Combinatorica 8, 63–74 (1988). https://doi.org/10.1007/BF02122553

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AMS subject classification (1980)

  • 05 C 15
  • 05 C 20
  • 05 C 38