On multiplicative graphs and the product conjecture Article Received: 01 November 1985 Revised: 21 January 1987 DOI:
Cite this article as: Häggkvist, R., Hell, P., Miller, D.J. et al. Combinatorica (1988) 8: 63. doi:10.1007/BF02122553 Abstract
We study the following problem: which graphs
G have the property that the class of all graphs not admitting a homomorphism into G 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 , cf also ) 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, . AMS subject classification (1980) 05 C 15 05 C 20 05 C 38
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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