On the Rainbow Vertex-Connection

Xueliang Li; Yongtang Shi

Discussiones Mathematicae Graph Theory (2013)

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

Abstract

top
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

top

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

top
  1. [1] N. Alon and J.H. Spencer, The Probabilistic Method, 3rd ed. (Wiley, New York, 2008). Zbl1148.05001
  2. [2] J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008). 
  3. [3] Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) R57. Zbl1181.05037
  4. [4] S. Chakraborty, E. Fischer, A. Matsliah and R. Yuster, Hardness and algorithms for rainbow connectivity, J. Comb. Optim. 21 (2011) 330-347. doi:10.1007/s10878-009-9250-9[Crossref] Zbl1319.05049
  5. [5] L. Chandran, A. Das, D. Rajendraprasad and N. Varma, Rainbow connection number and connected dominating sets, J. Graph Theory 71 (2012) 206-218. doi:10.1002/jgt.20643[Crossref] Zbl1248.05098
  6. [6] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohemica 133 (2008) 85-98. Zbl1199.05106
  7. [7] L. Chen, X. Li and Y. Shi, The complexity of determining the rainbow vertexconnection of a graph, Theoret. Comput. Sci. 412(35) (2011) 4531-4535. doi:10.1016/j.tcs.2011.04.032[Crossref] 
  8. [8] J.R. Griggs and M. Wu, Spanning trees in graphs with minimum degree 4 or 5, Discrete Math. 104 (1992) 167-183. doi:10.1016/0012-365X(92)90331-9[Crossref] 
  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

NotesEmbed ?

top

You must be logged in to post comments.