Multiplicativity of acyclic local tournaments

Abstract

A homomorphism of a digraph to another digraph is an edge-preserving vertex mapping. A digraphH is said to be multiplicative if the set of digraphs which do not admit a homomorphism toH is closed under categorical product. In this paper we discuss the multiplicativity of acyclic Hamiltonian digraphs, i.e., acyclic digraphs which contains a Hamiltonian path. As a consequence, we give a complete characterization of acyclic local tournaments with respect to multiplicativity.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    J. Bang-Jensen: Locally semicomplete digraphs: a generalization of tournaments,J. Graph Theory,14 (1990), 371–390.

    Google Scholar 

  2. [2]

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

    Google Scholar 

  3. [3]

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

    Google Scholar 

  4. [4]

    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 

  5. [5]

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

    Google Scholar 

  6. [6]

    A. Hajnal: The chromatic number of the product of two ℵ1 graphs can be countable,Combinatorica,5 (1985), 137–139.

    Google Scholar 

  7. [7]

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

  8. [8]

    P. Hell, H. Zhou, andX. Zhu: Homomorphisms to oriented cycles,Combinatorica,13 (1993), 421–433.

    Google Scholar 

  9. [9]

    P. Hell, H. Zhou, andX. Zhu: Multiplicativity of oriented cycles,Journal of Combinatorial Theory B,60, No. 2, (1994), 239–253.

    Google Scholar 

  10. [10]

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

    Google Scholar 

  11. [11]

    N. Sauer, andX. Zhu: An approach to Hedetniemi's conjecture,J. Graph Theory,16, 5 (1992), 423–436.

    Google Scholar 

  12. [12]

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

  13. [13]

    H. Zhou: Multiplicativity, part I—variations, multiplicative graphs and digraphs,J. Graph Theory,15 (1991), 469–488.

    Google Scholar 

  14. [14]

    H. Zhou: Multiplicativity, part II—non-multiplicative digraphs and characterization of oriented paths,J. Graph Theory,15 (1991), 489–509.

    Google Scholar 

  15. [15]

    H. Zhou: On the non-multiplicativity of oriented cycles,SIAM J. on Discrete Mathematics,5, No. 2 (1992), 207–218.

    Google Scholar 

  16. [16]

    H. Zhou: Multiplicativity of acyclic digraphs, to appear in Discrete Mathematics.

  17. [17]

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

  18. [18]

    X. Zhu: A simple proof of the multiplicativity of directed cycles of prime power length,Discrete Applied Mathematics,36 (1992), 313–315.

    Google Scholar 

  19. [19]

    X. Zhu: On the chromatic number of the products of hypergraphs,Ars Combinatorica.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zhou, H., Zhu, X. Multiplicativity of acyclic local tournaments. Combinatorica 17, 135–145 (1997). https://doi.org/10.1007/BF01196137

Download citation

Mathematics Subject Classification (1991)

  • 05 C 20