PDF
Discussiones Mathematicae Graph Theory 32(4) (2012)
783-793
DOI: https://doi.org/10.7151/dmgt.1644
On the Rainbow Connection of Cartesian Products and their Subgraphs
| Sandi Klavžar
Faculty of Mathematics and Physics, University of Ljubljana | Gašper Mekiš
Institute of Mathematics, Physics and Mechanics |
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.
References
| [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. |
Received 8 June 2011
Revised 6 February 2012
Accepted 6 February 2012
Close