Geodesic
Isomorphic components of direct products of bipartite graphs
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 231-248

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: direct product, tensor product, Kronecker product, bipartite graph 
Hammack, Richard. Isomorphic components of direct products of bipartite graphs. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 231-248. http://geodesic.mathdoc.fr/item/DMGT_2006_26_2_a5/
@article{DMGT_2006_26_2_a5,
     author = {Hammack, Richard},
     title = {Isomorphic components of direct products of bipartite graphs},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {231--248},
     year = {2006},
     volume = {26},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2006_26_2_a5/}
}
TY  - JOUR
AU  - Hammack, Richard
TI  - Isomorphic components of direct products of bipartite graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2006
SP  - 231
EP  - 248
VL  - 26
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/DMGT_2006_26_2_a5/
LA  - en
ID  - DMGT_2006_26_2_a5
ER  - 
%0 Journal Article
%A Hammack, Richard
%T Isomorphic components of direct products of bipartite graphs
%J Discussiones Mathematicae. Graph Theory
%D 2006
%P 231-248
%V 26
%N 2
%U http://geodesic.mathdoc.fr/item/DMGT_2006_26_2_a5/
%G en
%F DMGT_2006_26_2_a5

[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.

[2] W. Imrich and S. Klavžar, Product Graphs; Structure and Recognition (Wiley Interscience Series in Discrete Mathematics and Optimization, New York, 2000).

[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.

[4] P. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47-52, doi: 10.1090/S0002-9939-1962-0133816-6.