Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ ℕ, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex subset S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for each k-subset S of V (G) is called the k-rainbow index of G, denoted by rxk(G). In this paper, we first determine the graphs of size m whose 3-rainbow index equals m, m − 1, m − 2 or 2. We also obtain the exact values of rx3(G) when G is a regular multipartite complete graph or a wheel. Finally, we give a sharp upper bound for rx3(G) when G is 2-connected and 2-edge connected. Graphs G for which rx3(G) attains this upper bound are determined.
@article{bwmeta1.element.doi-10_7151_dmgt_1780,
author = {Lily Chen and Xueliang Li and Kang Yang and Yan Zhao},
title = {The 3-Rainbow Index of a Graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {81-94},
zbl = {1307.05066},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1780}
}
Lily Chen; Xueliang Li; Kang Yang; Yan Zhao. The 3-Rainbow Index of a Graph. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 81-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1780/
[1] J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008).
[2] Y. Caro, A. Lev, Y. Roditty, Zs. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15(1) (2008) R57. | Zbl 1181.05037
[3] G. Chartrand, G. Johns, K. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98. | Zbl 1199.05106
[4] G. Chartrand, F. Okamoto and P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55 (2010) 360-367. doi:10.1002/net.20339[WoS][Crossref] | Zbl 1205.05085
[5] G. Chartrand, G. Johns, K. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks 54(2) (2009) 75-81. doi:10.1002/net.20296[Crossref][WoS] | Zbl 1205.05124
[6] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, 2012). | Zbl 1250.05066
[7] X. Li, Y. Shi and Y. Sun, Rainbow connections of graphs: A survey, Graphs Combin. 29 (2013) 1-38. doi:10.1007/s00373-012-1243-2[WoS][Crossref] | Zbl 1258.05058