Vizing's conjecture: a survey and recent results

@article{Brear2012VizingsCA,
  title={Vizing's conjecture: a survey and recent results},
  author={Bo{\vs}tjan Bre{\vs}ar and Paul Dorbec and Wayne Goddard and Bert Hartnell and Michael A. Henning and Sandi Klav{\vz}ar and Douglas F. Rall},
  journal={Journal of Graph Theory},
  year={2012},
  volume={69}
}
Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary… 
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References

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TLDR
It is proved that if G and H are-regular then with only a few possible exceptions Vizing's conjecture holds and that for graphs of order at most n with minimum degrees at least p n ln n, the conjecture holds.
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TLDR
It is proved that all graphs G with a fair reception of size γ(G) satisfy Vizing’s conjecture on the domination number of Cartesian product graphs, by which the well-known result of Barcalkin and German concerning decomposable graphs is extended.
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TLDR
A new lower bound for the domination number of T T, when T is a tree, is established and an upper bound of Vizing is improved in the case when one of the graphs has k > 1 dominating sets which cover the vertex set and the other has a dominating set which partitions in a certain way.
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TLDR
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In this note we prove the following conjecture of Nowakowski and Rall: For arbitrary graphs G and H the upper domination number of the Cartesian product G H is at least the product of their upper
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TLDR
It is proved that the above equation has no nontrivial solution if H is one of the graphs obtained from Cn, the cycle of length n, either by adding a vertex and one pendant edge joining this vertex.
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TLDR
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TLDR
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On Vizing's conjecture
  • B. Brešar
  • Mathematics
    Discuss. Math. Graph Theory
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TLDR
A new concept which extends the ordinary domination of graphs is introduced, and it is proved that the conjecture Vizing's conjecture holds when γ(G) = γ (H) = 3, and the conjecture is generalized to Cartesian product G2H.
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