Discussiones Mathematicae Graph Theory 32(4) (2012) 783-793
doi: 10.7151/dmgt.1644

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On the Rainbow Connection of Cartesian Products and their Subgraphs

Sandi Klavžar

Faculty of Mathematics and Physics, University of Ljubljana
Jadranska 19, 1000 Ljubljana, Slovenia
and
Faculty of Natural Sciences and Mathematics, University of Maribor
Koroška 160, 2000 Maribor, Slovenia

Gašper Mekiš

Institute of Mathematics, Physics and Mechanics
Jadranska 19, 1000 Ljubljana

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.

Keywords: rainbow connection, strong rainbow connection, Cartesian product of graphs, isometric subgraph, hypercube

2010 Mathematics Subject Classification: 05C15, 05C76, 05C12.

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Received 8 June 2011
Revised 6 February 2012
Accepted 6 February 2012