Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, rc(G)⩽γc(G)+2, where γc(G) is the connected domination number of G. Bounds of the form diameter(G)⩽rc(G)⩽diameter(G)+c,1⩽c⩽4, for many special graph classes follow as easy corollaries from this result. We also show that every bridge-less chordal graph G has rc(G)⩽3⋅radius(G). An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree δ,rc(G)⩽3n/(δ+1)+3. This solves an open problem from Schiermeyer, 2009 [Schiermeyer, I. (2009). Rainbow connection in graphs with minimum degree three. In Fiala, J., Kratochvl, J., and Miller, M., editors, Combinatorial Algorithms, volume 5874 of Lecture Notes in Computer Science, pages 432-437. Springer Berlin / Heidelberg.], improving the previously best known bound of 20n/δ from Krivelevich and Yuster, 2010 [Krivelevich, M. and Yuster, R. (2010). The rainbow connection of a graph is (at most) reciprocal to its minimum degree. Journal of Graph Theory, 63(3):185-191.]. Moreover, this bound is seen to be tight up to additive factors by a construction mentioned in Caro et. al., 2008 [Caro, Y., Lev, A., Roditty, Y., Tuza, Z., and Yuster, R. (2008). On rainbow connection. The electronic journal of combinatorics, 15(R57):1].