A Note on the 3-rainbow Index of Complete Bipartite Graphs

A tree \( T \), in an edge-colored graph \( G \), is called a rainbow tree if no two edges of \( T \) are assigned the same color. For a vertex subset \( S \subseteq V(G) \), a tree that connects \( S \) in \( G \) is called an \( S \)-tree. A \( k \)-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 \( S \)-tree \( T \) in \( G \). The minimum number of colors needed in a \( k \)-rainbow coloring of \( G \) is the \( k \)-rainbow index of \( G \), denoted by \( rx_k(G) \). It is NP-hard to compute the \( rx_k(G) \) for the general graphs \( G \). We consider the \( 3 \)-rainbow index of complete bipartite graphs \( K_{s,t} \). For \( 3 \leq s \leq t \), we have determined the tight bounds of \( rx_3(K_{s,t}) \). In this paper, we continue the study. For \( 2 \leq s \leq t \), we develop a converse idea and apply it with the model of chessboard to study the problem. Finally, we obtain the exact value of \( rx_3(K_{s,t}) \) with \( 2 \leq s \leq t \).