A standard result states the direct product of two connected bipartite graphs has exactly two components. Jha, Klavžar and Zmazek proved that if one of the factors admits an automorphism that interchanges partite sets, then the components are isomorphic. They conjectured the converse to be true. We prove the converse holds if the factors are square-free. Further, we present a matrix-theoretic conjecture that, if proved, would prove the general case of the converse; if refuted, it would produce a counterexample.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1316,
author = {Richard Hammack},
title = {Isomorphic components of direct products of bipartite graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {26},
year = {2006},
pages = {231-248},
zbl = {1142.05055},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1316}
}
Richard Hammack. Isomorphic components of direct products of bipartite graphs. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 231-248. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1316/
[000] [1] T. Chow, The Q-spectrum and spanning trees of tensor products of bipartite graphs, Proc. Amer. Math. Soc. 125 (1997) 3155-3161, doi: 10.1090/S0002-9939-97-04049-5. | Zbl 0882.05089
[001] [2] W. Imrich and S. Klavžar, Product Graphs; Structure and Recognition (Wiley Interscience Series in Discrete Mathematics and Optimization, New York, 2000). | Zbl 0963.05002
[002] [3] P. Jha, S. Klavžar and B. Zmazek, Isomorphic components of Kronecker product of bipartite graphs, Discuss. Math. Graph Theory 17 (1997) 302-308, doi: 10.7151/dmgt.1057. | Zbl 0906.05050
[003] [4] P. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47-52, doi: 10.1090/S0002-9939-1962-0133816-6. | Zbl 0102.38801