Discussiones Mathematicae Graph Theory 30(4) (2010) 575-590
doi: 10.7151/dmgt.1515

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CANCELLATION OF DIRECT PRODUCTS OF DIGRAPHS

Richard H. Hammack  and  Katherine E. Toman

Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
Richmond, VA 23284-2014, USA
e-mail: rhammack@vcu.edu
e-mail: tomanke@vcu.edu

Abstract

We investigate expressions of form A×C ≅ B×C involving direct products of digraphs. Lovász gave exact conditions on C for which it necessarily follows that A ≅ B. We are here concerned with a different aspect of cancellation. We describe exact conditions on A for which it necessarily follows that A ≅ B. In the process, we do the following: Given an arbitrary digraph A and a digraph C that admits a homomorphism onto an arc, we classify all digraphs B for which A×C ≅ B×C.

Keywords: graph direct product, graph product cancellation, digraphs.

2010 Mathematics Subject Classification: Primary: 05C76; Secondary: 05C20, 05C60.

References

[1] R. Hammack, A cancellation property for the direct product of graphs, Discuss. Math. Graph Theory 28 (2008) 179-185, doi: 10.7151/dmgt.1400.
[2] R. Hammack, On direct product cancellation of graphs, Discrete Math. 309 (2009) 2538-2543, doi: 10.1016/j.disc.2008.06.004.
[3] P. Hell and J. Nesetril, Graphs and Homomorphisms, Oxford Lecture Series in Mathematics (Oxford U. Press, 2004), doi: 10.1093/acprof:oso/9780198528173.001.0001.
[4] W. Imrich and S. Klavžar, Product Graphs: Structure and recognition, Wiley-Interscience Series in Discrete Mathematics and Optimization (John Wiley and Sons, New York, 2000).
[5] L. Lovász, On the cancellation law among finite relational structures, Period. Math. Hungar. 1 (1971) 145-156, doi: 10.1007/BF02029172.

Received 16 July 2009
Revised 18 November 2009
Accepted 18 November 2009