The 3-rainbow index and connected dominating sets

A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of (G) is called a 3-rainbow coloring if for any three vertices in (G), there exists a rainbow tree connecting them. The 3-rainbow index (rx_3(G)) of (G) is defined as the minimum number of colors that are needed in a 3-rainbow coloring of (G). This concept, introduced by Chartrand et al., can be viewed as a generalization of the rainbow connection. In this paper, we study the 3-rainbow index by using connected 3-way dominating sets and 3-dominating sets. We show that for every connected graph (G) on (n) vertices with minimum degree at least (delta , (3le delta le 5)), (rx_{3}(G)le frac{3n}{delta +1}+4), and the bound is tight up to an additive constant; whereas for every connected graph (G) on (n) vertices with minimum degree at least (delta , (delta ge 3)), we get that (rx_{3}(G)le frac{ln (delta +1)}{delta +1}(1+o_{delta }(1))n+5). In addition, we obtain some tight upper bounds of the 3-rainbow index for some special graph classes, including threshold graphs, chain graphs and interval graphs.