# On the rainbow connection of Cartesian products and their subgraphs

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 4, page 783-793
- ISSN: 2083-5892

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topSandi Klavžar, and Gašper Mekiš. "On the rainbow connection of Cartesian products and their subgraphs." Discussiones Mathematicae Graph Theory 32.4 (2012): 783-793. <http://eudml.org/doc/270945>.

@article{SandiKlavžar2012,

abstract = {Rainbow connection number of Cartesian products and their subgraphs are considered. Previously known bounds are compared and non-existence of such bounds for subgraphs of products are discussed. It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above by the rainbow connection number of the hypercube. Isometric subgraphs of hypercubes with the rainbow connection number as small as possible compared to the rainbow connection of the hypercube are constructed. The concept of c-strong rainbow connected coloring is introduced. In particular, it is proved that the so-called Θ-coloring of an isometric subgraph of a hypercube is its unique optimal c-strong rainbow connected coloring.},

author = {Sandi Klavžar, Gašper Mekiš},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {rainbow connection; strong rainbow connection; Cartesian product of graphs; isometric subgraph; hypercube},

language = {eng},

number = {4},

pages = {783-793},

title = {On the rainbow connection of Cartesian products and their subgraphs},

url = {http://eudml.org/doc/270945},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Sandi Klavžar

AU - Gašper Mekiš

TI - On the rainbow connection of Cartesian products and their subgraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 4

SP - 783

EP - 793

AB - Rainbow connection number of Cartesian products and their subgraphs are considered. Previously known bounds are compared and non-existence of such bounds for subgraphs of products are discussed. It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above by the rainbow connection number of the hypercube. Isometric subgraphs of hypercubes with the rainbow connection number as small as possible compared to the rainbow connection of the hypercube are constructed. The concept of c-strong rainbow connected coloring is introduced. In particular, it is proved that the so-called Θ-coloring of an isometric subgraph of a hypercube is its unique optimal c-strong rainbow connected coloring.

LA - eng

KW - rainbow connection; strong rainbow connection; Cartesian product of graphs; isometric subgraph; hypercube

UR - http://eudml.org/doc/270945

ER -

## References

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- [9] M. Krivelevich and R. Yuster, The rainbow connection of a graph is (at most) reciprocal to its minimum degree, J. Graph Theory 63 (2010) 185-191, doi: 10.1002/jgt.20418. Zbl1193.05079
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