Note on the upper bound of the rainbow index of a graph

Abstract

A path in an edge-colored graph G 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 is 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 containing all the vertices of S (other vertices may also be included in the tree) 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 number, 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 Szemerédi’s Regularity Lemma, connected 2-step dominating sets, connected (k−1)-dominating sets and k-dominating sets of G.