NDLI: Rainbow Connection Number and Radius

Content Provider	Springer Nature Link
Author	Basavaraju, Manu Chandran, L. Sunil Rajendraprasad, Deepak Ramaswamy, Arunselvan
Copyright Year	2012
Abstract	The 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 its 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. in Electron J Comb 15(1):Research paper 57, 13, 2008). It is known that computing rc(G) is NP-Hard (Chakraborty and fischer in J Comb Optim 1–18, 2009). Here, we present a (r + 3)-factor approximation algorithm which runs in O(nm) time and a (d + 3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.
Starting Page	275
Ending Page	285
Page Count	11
File Format	PDF
ISSN	09110119
Journal	Graphs and Combinatorics
Volume Number	30
Issue Number	2
e-ISSN	14355914
Language	English
Publisher	Springer Japan
Publisher Date	2012-11-29
Publisher Place	Tokyo
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Rainbow connectivity Rainbow colouring Radius Isometric cycle Chordality Approximation algorithm Combinatorics Engineering Design
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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