Upper bounding rainbow connection number by forest number

A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and rainbow-connected there a between each pair its vertices. The minimum number colors needed to rainbow-connect G connection G, denoted by rc(G). simple way color spanning tree with distinct then re-use any these remaining G. This proves that rc(G)≤|V(G)|−1. We ask whether stronger tree-like structures coloring than implied above trivial argument. For instance, possible find upper bound t(G)−1 for rc(G), where t(G) vertices largest induced G? answer turns out be negative, as counter-examples show even c⋅t(G) not rc(G) given constant c. In this work we consider forest f(G), maximum instead t(G), surprisingly do get bound. More specifically, prove rc(G)≤f(G)+2. Our result indicates was suggested based