The 3-rainbow index and connected dominating sets

Abstract: 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 3-rainbow 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 three-way dominating sets and 3-dominating sets. We shown that for every connected graph $G$ on $n$ vertices with minimum degree at least $\delta$ ($3\leq\delta\leq5$), $rx_{3}(G)\leq \frac{3n}{\delta+1}+4$, and the bound is tight up to an additive constant; whereas for every connected graph $G$ on $n$ vertices with minimum degree at least $\delta$ ($\delta\geq3$), we get that $rx_{3}(G)\leq n\frac{ln(\delta+1)}{\delta+1}(1+o_{\delta}(1))+5$. 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.