Discussiones Mathematicae Graph Theory  17(2) (1997)  301-309
doi: 10.7151/dmgt.1057

ISOMORPHIC COMPONENTS OF KRONECKER PRODUCT OF BIPARTITE GRAPHS

 Pranava K. Jha Department of Computer Engineering Delhi Institute of Technology: Delhi Kashmere Gate Delhi 110 006, India e-mail: pkj@dit.ernet.in Sandi Klavžar and Blaž Zmazek Department of Mathematics, PEF, University of Maribor Koro ska cesta 160, 2000 Maribor, Slovenia e-mail: sandi.klavzar@uni-lj.si

## Abstract

Weichsel (Proc. Amer. Math. Soc. 13 (1962) 47-52) proved that the Kronecker product of two connected bipartite graphs consists of two connected components. A condition on the factor graphs is presented which ensures that such components are isomorphic. It is demonstrated that several familiar and easily constructible graphs are amenable to that condition. A partial converse is proved for the above condition and it is conjectured that the converse is true in general.

Keywords: Kronecker product, bipartite graphs, graph isomorphism.

1991 Mathematics Subject Classification: 05C60.

## References

 [1] L. Babai, Automorphism Groups, Isomorphism, Reconstruction, Chapter 27 in Handbook of Combinatorics (R.L. Graham. M. Grötschel, L. Lovász, eds.) Elsevier, Amsterdam, (1995) 1447-1540. [2] D. Greenwell and L. Lovász, Applications of product colouring, Acta Math. Acad. Sci. Hungar. 25 (1974) 335-340, doi: 10.1007/BF01886093. [3] S. Hedetniemi, Homomorphisms of graphs and automata, University of Michigan, Technical Report 03105-44-T, (1966). [4] P. Hell, An introduction to the category of graphs, Ann. New York Acad. Sci. 328 (1979) 120-136, doi: 10.1111/j.1749-6632.1979.tb17773.x. [5] W. Imrich, Factorizing cardinal product graphs in polynomial time, Discrete Math., to appear. [6] P.K. Jha, Decompositions of the Kronecker product of a cycle and a path into long cycles and long paths, Indian J. Pure Appl. Math. 23 (1992) 585-602. [7] P.K. Jha, Hamiltonian decompositions of products of cycles, Indian J. Pure Appl. Math. 23 (1992) 723-729. [8] P.K. Jha, N. Agnihotri and R. Kumar, Edge exchanges in Hamiltonian decompositions of Kronecker-product graphs, Comput. Math. Applic. 31 (1996) 11-19, doi: 10.1016/0898-1221(95)00189-1. [9] F. Lalonde, Le probleme d'etoiles pour graphes est NP-complet, Discrete Math. 33 (1981) 271-280, doi: 10.1016/0012-365X(81)90271-5. [10] R.H. Lamprey and B.H. Barnes, Product graphs and their applications, Modelling and Simulation, 5 (1974) 1119-1123 (Proc Fifth Annual Pittsburgh Conference, Instrument Society of America, Pittsburgh, PA, 1974). [11] J. Neetil, Representations of graphs by means of products and their complexity, Lecture Notes in Comput. Sci. 118 (1981) 94-102, doi: 10.1007/3-540-10856-4_76. [12] E. Pesch, Minimal extensions of graphs to absolute retracts, J. Graph Theory 11 (1987) 585-598, doi: 10.1002/jgt.3190110416. [13] V.G. Vizing, The Cartesian product of graphs, Vyc. Sis. 9 (1963) 30-43. [14] P.M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47-52, doi: 10.1090/S0002-9939-1962-0133816-6.