Discussiones Mathematicae Graph Theory 33(2) (2013) 307-313
doi: 10.7151/dmgt.1664

[BIBTex] [PDF] [PS]

On the Rainbow Vertex-connection

Xueliang Li and Yongtang Shi

Center for Combinatorics and LPMC-TJKLC
Nankai University, Tianjin 300071, China

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.

References

[1]N. Alon and J.H. Spencer, The Probabilistic Method, 3rd ed. (Wiley, New York, 2008).
[2]J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008).
[3]Y. Caro, A. Lev, Y. Roditty, Z. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) R57.
[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 .
[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.
[6]G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohemica 133 (2008) 85--98.
[7]L. Chen, X. Li and Y. Shi, The complexity of determining the rainbow vertex-connection of a graph, Theoret. Comput. Sci. 412(35) (2011) 4531--4535, doi: 10.1016/j.tcs.2011.04.032.
[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.
[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.
[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.
[11]X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, New York, 2012).
[12]N. Linial and D. Sturtevant, Unpublished result (1987).
[13]I. Schiermeyer, Rainbow connection in graphs with minimum degree three, IWOCA 2009, LNCS 5874 (2009) 432--437.

Received 27 June 2011
Revised 5 March 2012
Accepted 5 March 2012