# On the Rainbow Vertex-Connection

• Volume: 33, Issue: 2, page 307-313
• ISSN: 2083-5892

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## Abstract

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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 vertexconnected. 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 [xxx] and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5 for [xxx] and rvc(G) ≤ 4n/(δ + 1) + C(δ) for 6 ≤ δ ≤ 15, where [xxx]. 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 [xxx] and δ = 3, 4, 5, our bounds are seen to be tight up to additive constants.

## How to cite

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Xueliang Li, and Yongtang Shi. "On the Rainbow Vertex-Connection." Discussiones Mathematicae Graph Theory 33.2 (2013): 307-313. <http://eudml.org/doc/268108>.

@article{XueliangLi2013,
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 vertexconnected. 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 [xxx] and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5 for [xxx] and rvc(G) ≤ 4n/(δ + 1) + C(δ) for 6 ≤ δ ≤ 15, where [xxx]. 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 [xxx] and δ = 3, 4, 5, our bounds are seen to be tight up to additive constants.},
author = {Xueliang Li, Yongtang Shi},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {rainbow vertex-connection; vertex coloring; minimum degree; 2-step dominating set},
language = {eng},
number = {2},
pages = {307-313},
title = {On the Rainbow Vertex-Connection},
url = {http://eudml.org/doc/268108},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Xueliang Li
AU - Yongtang Shi
TI - On the Rainbow Vertex-Connection
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 2
SP - 307
EP - 313
AB - 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 vertexconnected. 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 [xxx] and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5 for [xxx] and rvc(G) ≤ 4n/(δ + 1) + C(δ) for 6 ≤ δ ≤ 15, where [xxx]. 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 [xxx] and δ = 3, 4, 5, our bounds are seen to be tight up to additive constants.
LA - eng
KW - rainbow vertex-connection; vertex coloring; minimum degree; 2-step dominating set
UR - http://eudml.org/doc/268108
ER -

## References

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9. [9] D.J. Kleitman and D.B. West, Spanning trees with many leaves, SIAM J. Discrete Math. 4 (1991) 99-106. doi:10.1137/0404010[WoS][Crossref] Zbl0734.05041
10. [10] M. Krivelevich and R. Yuster, The rainbow connection of a graph is (at most) reciprocal to its minimum degree, J. Graph Theory 63 (2010) 185-191. doi:/10.1002/jgt.20418[WoS] Zbl1193.05079
11. [11] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, New York, 2012). Zbl1250.05066
12. [12] N. Linial and D. Sturtevant, Unpublished result (1987).
13. [13] I. Schiermeyer, Rainbow connection in graphs with minimum degree three, IWOCA 2009, LNCS 5874 (2009) 432-437. Zbl1267.05125

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