Abstract
A tree in an edge-colored graph G is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers k, \(\ell \) with \(k\ge 3\), the \((k,\ell )\) -rainbow index \(rx_{k,\ell }(G)\) of G is the minimum number of colors needed in an edge-coloring of G such that for any set S of k vertices of G, there exist \(\ell \) internally disjoint rainbow trees connecting S. This concept was introduced by Chartrand et. al., and there have been very few known results about it. In this paper, we establish a sharp threshold function for \(rx_{k,\ell }(G_{n,p})\le k\) and \(rx_{k,\ell }(G_{n,M})\le k,\) respectively, where \(G_{n,p}\) and \(G_{n,M}\) are the usually defined random graphs.
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References
Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley, New York (2004)
Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001)
Bondy, J.A., Murty, U.S.R.: Graph Theory, GTM 244. Springer, New York (2008)
Cai, Q., Li, X., Song, J.: Solutions to conjectures on the \((k,\ell )\)-rainbow index of complete graphs. Networks 62, 220–224 (2013)
Caro, Y., Lev, A., Roditty, Y., Tuza, Z., Yuster, R.: On rainbow connection. Electron. J. Combin. 15(1), R57 (2008)
Chandran, L., Das, A., Rajendraprasad, D., Varma, N.: Rainbow connection number and connected dominating sets. J. Graph Theory 71(2), 206–218 (2012)
Chartrand, G., Johns, G., McKeon, K., Zhang, P.: Rainbow connection in graphs. Math. Bohem. 133, 85–98 (2008)
Chartrand, G., Johns, G., McKeon, K., Zhang, P.: The rainbow connectivity of a graph. Networks 54(2), 75–81 (2009)
Chartrand, G., Okamoto, F., Zhang, P.: Rainbow trees in graphs and generalized connectivity. Networks 55, 360–367 (2010)
Fujita, S., Liu, H., Magnant, C.: Rainbow k-connection in dense graphs. Electron. Notes Discrete Math. 38, 361-366 (2011), or, J. Combin. Math. Combin. Comput., to appear
Huang, X., Li, X., Shi, Y.: Note on the hardness of rainbow connections for planar and line graphs. Bull. Malays. Math. Sci. Soc. 38(3), 1235–1241 (2015)
Krivelevich, M., Yuster, R.: The rainbow connection of a graph is (at most) reciprocal to its minimum degree. J. Graph Theory 63(3), 185–191 (2010)
Li, X., Sun, Y.: Rainbow Connections of Graphs. Springer Briefs in Math., Springer, New York (2012)
Li, X., Sun, Y.: On the strong rainbow connection of a graph. Bull. Malays. Math. Sci. Soc. (2) 36(2), 299–311 (2013)
Li, S., Li, W., Li, X.: The generalized connectivity of complete bipartite graphs. Ars Combin. 104, 65–79 (2012)
Li, X., Shi, Y., Sun, Y.: Rainbow connections of graphs: a survey. Graphs Combin. 29(1), 1–38 (2013)
Li, S., Li, W., Li, X.: The generalized connectivity of complete equipartition 3-partite graphs. Bull. Malays. Math. Sci. Soc. (2) 37(1), 103–121 (2014)
Li, H., Li, X., Mao, Y.: On extremal graphs with at most two internally disjoint Steiner trees connecting any three vertices. Bull. Malays. Math. Sci. Soc. (2), in press
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The authors are very grateful to the reviewers for their helpful comments and suggestions. Supported by NSFC No. 11371205 and 11071130.
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Communicated by Sanming Zhou.
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Cai, Q., Li, X. & Song, J. The \((k,\ell )\)-Rainbow Index of Random Graphs. Bull. Malays. Math. Sci. Soc. 39, 765–771 (2016). https://doi.org/10.1007/s40840-015-0301-3
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DOI: https://doi.org/10.1007/s40840-015-0301-3