A path in an edge-colored graph G, where adjacent edges may be colored the
same, is a rainbow path if every two edges of it receive distinct colors. The
rainbow connection number of a connected graph G, denoted by rc(G), is the
minimum number of colors that are needed to color the edges of G such that
there exists a rainbow path connecting every two vertices of G. Similarly, a
tree in G is a rainbow~tree if no two edges of it receive the same color. The
minimum number of colors that are needed in an edge-coloring of G such that
there is a rainbow tree connecting S for each k-subset S of V(G) is
called the k-rainbow index of G, denoted by rxk​(G), where k is an
integer such that 2≤k≤n. Chakraborty et al. got the following result:
For every ϵ>0, a connected graph with minimum degree at least
ϵn has bounded rainbow connection, where the bound depends only on
ϵ. Krivelevich and Yuster proved that if G has n vertices and the
minimum degree δ(G) then rc(G)<20n/δ(G). This bound was later
improved to 3n/(δ(G)+1)+3 by Chandran et al. Since rc(G)=rx2​(G), a
natural problem arises: for a general k determining the true behavior of
rxk​(G) as a function of the minimum degree δ(G). In this paper, we
give upper bounds of rxk​(G) in terms of the minimum degree δ(G) in
different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected
2-step dominating sets, connected (k−1)-dominating sets and k-dominating
sets of G.Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by
other author