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Discussiones Mathematicae Graph Theory 33(2) (2013)
307-313
DOI: https://doi.org/10.7151/dmgt.1664
On the Rainbow Vertex-connection
| Xueliang Li and Yongtang Shi
Center for Combinatorics and LPMC-TJKLC |
Abstract
A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. It was proved that if G is a graph of order n with minimum degree δ, then rvc(G) < 11n/ δ. In this paper, we show that rvc(G) ≤ 3n/( δ+1)+5 for δ ≥ √n −1 −1 and n ≥ 290, while rvc(G) ≤ 4n/( δ+1)+5 for 16 ≤ δ ≤ √n −1 −2 and rvc(G) ≤ 4n/( δ+1)+C( δ) for 6 ≤ δ ≤ 15, where C( δ) = e(3log( δ3+2 δ2+3) −3(log3 −1))/(δ −3) −2. We also prove that rvc(G) ≤ 3n/4 −2 for δ = 3, rvc(G) ≤ 3n/5 −8/5 for δ = 4 and rvc(G) ≤ n/2 −2 for δ = 5. Moreover, an example constructed by Caro et al. shows that when δ ≥ √n −1 −1 and δ = 3,4,5, our bounds are seen to be tight up to additive constants.
Keywords: rainbow vertex-connection, vertex coloring, minimum degree, 2-step dominating set
2010 Mathematics Subject Classification: 05C15, 05C40.
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Received 27 June 2011
Revised 5 March 2012
Accepted 5 March 2012
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