Rainbow Trees in Small Cubic Graphs

Let \( G \) be a nontrivial connected graph of order \( n \) and \( k \) an integer with \( 2 \leq k \leq n \). For a set \( S \) of \( k \) vertices of \( G \), let \( \kappa(S) \) denote the maximum number \( \ell \) of pairwise edge-disjoint trees \( T_1, T_2, \ldots, T_\ell \) in \( G \) such that \( V(T_i) \cap V(T_j) = S \) for every pair \( i, j \) of distinct integers with \( 1 \leq i, j \leq \ell \). A collection \( \{T_1, T_2, \ldots, T_\ell\} \) of trees in \( G \) with this property is called a set of internally disjoint trees connecting \( S \). The \( k \)-connectivity \( \kappa_k(G) \) of \( G \) is defined as \( \kappa_k(G) = \text{min}\{\kappa(S)\} \), where the minimum is taken over all \( k \)-element subsets \( S \) of \( V(G) \). Thus \( \kappa_2(G) \) is the connectivity \( \kappa(G) \) of \( G \). In an edge-colored graph \( G \) in which adjacent edges may be colored the same, a tree \( T \) is a rainbow tree in \( G \) if no two edges of \( T \) are colored the same. For each integer \( \ell \) with \( 1 \leq \ell \leq \kappa_k(G) \), a \( (k, \ell) \)-rainbow coloring of \( G \) is an edge coloring of \( G \) (in which adjacent