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 the graph is rainbow-connected if there is a rainbow path between each pair of its vertices. The minimum number of colors needed to rainbow-connect a graph G is the rainbow connection number of G, denoted by rc(G). A simple way to rainbow-connect a graph G is to color the edges of a spanning tree with distinct colors and then re-use any of these colors to color the remaining edges of G. This proves that rc(G)≤|V(G)|−1. We ask whether there is a stronger connection between tree-like structures and rainbow coloring than that is implied by the above trivial argument. For instance, is it possible to find an upper bound of t(G)−1 for rc(G), where t(G) is the number of vertices in the largest induced tree of G? The answer turns out to be negative, as there are counter-examples that show that even c⋅t(G) is not an upper bound for rc(G) for any given constant c. In this work we show that if we consider the forest number f(G), the number of vertices in a maximum induced forest of G, instead of t(G), then surprisingly we do get an upper bound. More specifically, we prove that rc(G)≤f(G)+2. Our result indicates a stronger connection between rainbow connection and tree-like structures than that was suggested by the simple spanning tree based upper bound.