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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1586,
author = {Futaba Fujie-Okamoto and Kyle Kolasinski and Jianwei Lin and Ping Zhang},
title = {Vertex rainbow colorings of graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {32},
year = {2012},
pages = {63-80},
zbl = {1255.05080},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1586}
}
Futaba Fujie-Okamoto; Kyle Kolasinski; Jianwei Lin; Ping Zhang. Vertex rainbow colorings of graphs. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 63-80. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1586/
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