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Discussiones Mathematicae Graph Theory 28(1) (2008)
179-184
DOI: https://doi.org/10.7151/dmgt.1400
A CANCELLATION PROPERTY FOR THE DIRECT PRODUCT OF GRAPHS
Richard H. Hammack
Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
Richmond, VA 23284-2014, USA
e-mail: rhammack@vcu.edu
Abstract
Given graphs A, B and C for which A×C ≅ B×C, it is not generally true that A ≅ B. However, it is known that A×C ≅ B×C implies A ≅ B provided that C is non-bipartite, or that there are homomorphisms from A and B to C. This note proves an additional cancellation property. We show that if B and C are bipartite, then A×C ≅ B×C implies A ≅ B if and only if no component of B admits an involution that interchanges its partite sets.Keywords: graph products, graph direct product, cancellation.
2000 Mathematics Subject Classification: 05C60.
References
[1] | R. Hammack, A quasi cancellation property for the direct product, Proceedings of the Sixth Slovenian International Conference on Graph Theory, under review. |
[2] | 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. |
[3] | W. Imrich and S. Klavžar, Product Graphs; Structure and Recognition (Wiley Interscience Series in Discrete Mathematics and Optimization, 2000). |
[4] | L. Lovász, On the cancellation law among finite relational structures, Period. Math. Hungar. 1 (1971) 59-101. |
Received 13 August 2007
Revised 5 December 2007
Accepted 5 December 2007
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