Vertex rainbow colorings of graphs

In a properly vertex-colored graphG, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P . If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring ofG. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number vrc(G) of G. Thus if G is a connected graph of order n ≥ 2, then 2 ≤ vrc(G) ≤ n. We present characterizations of all connected graphs G of order n for which vrc(G) ∈ {2, n−1, n} and study the relationship between vrc(G) and the chromatic number χ(G) of G. For a connected graph G of order n and size m, the number m − n + 1 is the cycle rank of G. Vertex rainbow connection numbers are determined for all connected graphs of cycle rank 0 or 1 and these numbers are investigated for connected graphs of cycle rank 2.