, 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


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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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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