Rainbow connection numbers of line graphs, middle graphs and total graphs

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph G is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of G, denoted rcG), is the minimum number of colors that are needed in order to make G rainbow connected. In this paper, we investigate the rainbow connection numbers of the line graph, middle graph and total graph of a connected triangle-free graph G. We obtain three near) sharp upper bounds in terms of the number of vertex-disjoint cycles of the original graph G.