Basavaraju, Manu and Chandran, Sunil L and Ramaswamy, Arunselvan and Rajendraprasad, Deepak
(2010)
*Rainbow Connection Number and Radius.*
UNSPECIFIED.

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## Abstract

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r+2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that, for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (K_{1,n} for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4n/5 - 1, where n is order of the graph [Caro et al., 2008]

Item Type: | Departmental Technical Report |
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Keywords: | Rainbow connectivity;rainbow colouring;radius;isometric cycle;chordality. |

Department/Centre: | Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) |

Date Deposited: | 09 Nov 2011 06:30 |

Last Modified: | 09 Nov 2011 06:30 |

URI: | http://eprints.iisc.ernet.in/id/eprint/40344 |

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