Menu
A tree \( T \), in an edge-colored graph \( G \), is called a \emph{rainbow tree} if no two edges of \( T \) are assigned the same color. A \( k \)-\emph{rainbow coloring} of \( G \) is an edge coloring of \( G \) having the property that for every set \( S \) of \( k \) vertices of \( G \), there exists a rainbow tree \( T \) in \( G \) such that \( S \subseteq V(T) \). The minimum number of colors needed in a \( k \)-rainbow coloring of \( G \) is the \( k \)-\emph{rainbow index} of \( G \), denoted by \( \text{rx}_k(G) \). In this paper, we investigate the 3-rainbow index \( \text{rx}_3(G) \) of a connected graph \( G \). For a connected graph \( G \), it is shown that a sharp upper bound of \( \text{rx}_3(G) \) is \( \text{rx}_3(G[D]) + 4 \), where \( D \) is a connected 3-way dominating set and a connected 2-dominating set of \( G \). Moreover, we determine a sharp upper bound for \( K_{s,t} \) (\( 3 \leq s \leq t \)) and a better bound for \((P_5,C_5)\)-free graphs, respectively. Finally, a sharp bound for the 3-rainbow index of general graphs is obtained.