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2013 | 33 | 2 | 307-313
Tytuł artykułu

On the Rainbow Vertex-Connection

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
33
Numer
2
Strony
307-313
Opis fizyczny
Daty
wydano
2013-05-01
online
2013-04-13
Twórcy
autor
  • Center for Combinatorics and LPMC-TJKLC Nankai University, Tianjin 300071, China, lxl@nankai.edu.cn
autor
  • Center for Combinatorics and LPMC-TJKLC Nankai University, Tianjin 300071, China, shi@nankai.edu.cn
Bibliografia
  • [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[Crossref]
  • [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]
  • [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 vertexconnection of a graph, Theoret. Comput. Sci. 412(35) (2011) 4531-4535. doi:10.1016/j.tcs.2011.04.032[Crossref]
  • [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] 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]
  • [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]
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1664
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