Isomorphic components of direct products of bipartite graphs

Richard Hammack

Discussiones Mathematicae Graph Theory (2006)

  • Volume: 26, Issue: 2, page 231-248
  • ISSN: 2083-5892

Abstract

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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.

How to cite

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Richard Hammack. "Isomorphic components of direct products of bipartite graphs." Discussiones Mathematicae Graph Theory 26.2 (2006): 231-248. <http://eudml.org/doc/270238>.

@article{RichardHammack2006,
abstract = {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.},
author = {Richard Hammack},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {direct product; tensor product; Kronecker product; bipartite graph},
language = {eng},
number = {2},
pages = {231-248},
title = {Isomorphic components of direct products of bipartite graphs},
url = {http://eudml.org/doc/270238},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Richard Hammack
TI - Isomorphic components of direct products of bipartite graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 2
SP - 231
EP - 248
AB - 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.
LA - eng
KW - direct product; tensor product; Kronecker product; bipartite graph
UR - http://eudml.org/doc/270238
ER -

References

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  1. [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. Zbl0882.05089
  2. [2] W. Imrich and S. Klavžar, Product Graphs; Structure and Recognition (Wiley Interscience Series in Discrete Mathematics and Optimization, New York, 2000). Zbl0963.05002
  3. [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. Zbl0906.05050
  4. [4] P. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47-52, doi: 10.1090/S0002-9939-1962-0133816-6. Zbl0102.38801

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