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.
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Caro, Y., Lev, A., Roditty, Y., Tuza, Z., Yuster, R.: On rainbow connection. Electron. J. Comb. 15(1), Research paper 57, 13 (2008)
Chakraborty, S., Fischer, E., Matsliah, A., Yuster, R.: Hardness and algorithms for rainbow connection. J. Comb. Optim. 1–18 (2009)
Chandran L.S., Das A., Rajendraprasad D., Varma N.M.: Rainbow connection number and connected dominating sets. J. Graph Theory 21, 330–347 (2011)
Chartrand G., Johns G.L., McKeon K.A., Zhang P.: Rainbow connection in graphs. Math. Bohem 133(1), 85–98 (2008)
Chartrand G., Zhang P.: Chromatic graph theory. Chapman & Hall, London (2008)
Dong, J., Li, X.: Rainbow connection number, bridges and radius. Preprint arXiv:1105.0790v1 [math.CO] (2011)
Krivelevich M., Yuster R.: The rainbow connection of a graph is at most reciprocal to its minimum degree. J. Graph Theory 63(3), 185–191 (2010)
Li X., Liu S., Chandran L.S., Mathew R., Rajendraprasad D.: Rainbow connection number and connectivity. Electron. J. Comb. 19(1), 20 (2012)
Li, X., Shi Y.: Rainbow connection in 3-connected graphs. Graphs Comb. 1–5 (2012). doi:10.1007/s00373-012-1204-9
Tarjan R.E.: A note on finding the bridges of a graph. Inf. Process. Lett. 2(6), 160–161 (1974)
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Partially supported by Microsoft Research India–Ph.D. Fellowship.
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Basavaraju, M., Chandran, L.S., Rajendraprasad, D. et al. Rainbow Connection Number and Radius. Graphs and Combinatorics 30, 275–285 (2014). https://doi.org/10.1007/s00373-012-1267-7
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DOI: https://doi.org/10.1007/s00373-012-1267-7