A note on the 3-rainbow index of $K_{2,t}$

TL;DR

This paper determines the exact 3-rainbow index for the complete bipartite graph $K_{2,t}$ for all $t \

Contribution

It provides the first exact values of the 3-rainbow index for the class of graphs $K_{2,t}$, filling a gap in rainbow connectivity research.

Findings

Exact values of $rx_3(K_{2,t})$ for all $t \\geq 1$ are obtained.

The paper characterizes the minimal colorings needed for 3-rainbow connectivity.

Results contribute to understanding rainbow indices of bipartite graphs.

Abstract

A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. For a vertex subset $S\in V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. A {\em $k$-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 {\em $k$-rainbow index of $G$}, denoted by $rx_k(G)$. In this paper, we obtain the exact values of $rx_3(K_{2,t})$ for any $t\geq 1$.