Geodesic
On the rainbow connection of Cartesian products and their subgraphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 783-793.

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

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 
@article{DMGT_2012_32_4_a12,
     author = {Klav\v{z}ar, Sandi and Meki\v{s}, Ga\v{s}per},
     title = {On the rainbow connection of {Cartesian} products and their subgraphs},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {783--793},
     publisher = {mathdoc},
     volume = {32},
     number = {4},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a12/}
}
TY  - JOUR
AU  - Klavžar, Sandi
AU  - Mekiš, Gašper
TI  - On the rainbow connection of Cartesian products and their subgraphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2012
SP  - 783
EP  - 793
VL  - 32
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a12/
LA  - en
ID  - DMGT_2012_32_4_a12
ER  - 
%0 Journal Article
%A Klavžar, Sandi
%A Mekiš, Gašper
%T On the rainbow connection of Cartesian products and their subgraphs
%J Discussiones Mathematicae. Graph Theory
%D 2012
%P 783-793
%V 32
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a12/
%G en
%F DMGT_2012_32_4_a12
Klavžar, Sandi; Mekiš, Gašper. On the rainbow connection of Cartesian products and their subgraphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 783-793. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a12/

[1] M. Basavaraju, L.S. Chandran, D. Rajendraprasad and A. Ramaswamy, Rainbow connection number of graph power and graph products, manuscript (2011) arXiv:1104.4190 [math.CO].

[2] Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) #R57.

[3] S. Chakraborty, E. Fischer, A. Matsliah and R. Yuster, Hardness and algorithms for rainbow connection, J. Comb. Optim. 21 (2011) 330-347, doi: 10.1007/s10878-009-9250-9.

[4] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98.

[5] T. Gologranc, G. Mekiš and I. Peterin, Rainbow connection and graph products, IMFM Preprint Series 49 (2011) #1149.

[6] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs: Second Edition (CRC Press, Boca Raton, 2011).

[7] W. Imrich, S. Klavžar and D.F. Rall, Topics in Graph Theory: Graphs and Their Cartesian Products (A K Peters, Wellesley, 2008).

[8] A. Kemnitz and I. Schiermeyer, Graphs with rainbow connection number two, Discuss. Math. Graph Theory 31 (2011) 313-320, doi: 10.7151/dmgt.1547.

[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.

[10] X. Li and Y. Sun, Characterize graphs with rainbow connection number m-2 and rainbow connection numbers of some graph operations, manuscript (2010).

[11] X. Li and Y. Sun, Rainbow connection of graphs - A survey, manuscript (2011) arXiv:1101.5747v1 [math.CO].

[12] I. Schiermeyer, Bounds for the rainbow connection number of graphs, Discuss. Math. Graph Theory 31 (2011) 387-395, doi: 10.7151/dmgt.1553.

[13] P. Winkler, Isometric embedding in products of complete graphs, Discrete Appl. Math. 7 (1984) 221-225.