Geodesic
Vertex rainbow colorings of graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 63-80

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In a properly vertex-colored graph G, 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 of G. 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.
Keywords: rainbow path, vertex rainbow coloring, vertex rainbow connection number 
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Fujie-Okamoto, Futaba; Kolasinski, Kyle; Lin, Jianwei; Zhang, Ping. Vertex rainbow colorings of graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 63-80. http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a5/