The rainbow connection of a graph is (at most) reciprocal to its minimum degree

An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edge-connected. We prove that if G has n vertices and minimum degreeδ then rc(G)<20n /δ. This solves open problems from Y. Caro, A. Lev Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), #R57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243-254). A vertex-colored graph G is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if G has n vertices and minimum degree δthen rvc(G)<11n /δ We note that the proof in this case is different from the proof for the edge-colored case, and we cannot deduce one from the other.