Geodesic
The 3-Rainbow Index of a Graph
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 81-94 Cet article a éte moissonné depuis la source Library of Science

Voir la notice de l'article

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 rx_k(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 rx_3(G) when G is a regular multipartite complete graph or a wheel. Finally, we give a sharp upper bound for rx_3(G) when G is 2-connected and 2-edge connected. Graphs G for which rx_3(G) attains this upper bound are determined.
Keywords: rainbow tree, S-tree, k-rainbow index 
@article{DMGT_2015_35_1_a6,
     author = {Chen, Lily and Li, Xueliang and Yang, Kang and Zhao, Yan},
     title = {The {3-Rainbow} {Index} of a {Graph}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {81--94},
     year = {2015},
     volume = {35},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a6/}
}
TY  - JOUR
AU  - Chen, Lily
AU  - Li, Xueliang
AU  - Yang, Kang
AU  - Zhao, Yan
TI  - The 3-Rainbow Index of a Graph
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2015
SP  - 81
EP  - 94
VL  - 35
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a6/
LA  - en
ID  - DMGT_2015_35_1_a6
ER  - 
%0 Journal Article
%A Chen, Lily
%A Li, Xueliang
%A Yang, Kang
%A Zhao, Yan
%T The 3-Rainbow Index of a Graph
%J Discussiones Mathematicae. Graph Theory
%D 2015
%P 81-94
%V 35
%N 1
%U http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a6/
%G en
%F DMGT_2015_35_1_a6
Chen, Lily; Li, Xueliang; Yang, Kang; Zhao, Yan. The 3-Rainbow Index of a Graph. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 81-94. http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a6/

[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.

[3] G. Chartrand, G. Johns, K. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98.

[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

[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

[6] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, 2012).

[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