Discussiones Mathematicae Graph Theory 32(2) (2012) 63-80
doi: 10.7151/dmgt.1586

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Vertex Rainbow Colorings of Graphs

Futaba Fujie-Okamoto

Mathematics Department
University of Wisconsin La Crosse
La Crosse, WI 54601, USA

Kyle Kolasinski, Jianwei Lin, Ping Zhang

Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008, USA


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

2010 Mathematics Subject Classification: 05C15, 05C40.


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Received 15 June 2010
Revised 20 January 2011
Accepted 24 January 2011