The strong 3-rainbow index of edge-comb product of a path and a connected graph

Abstract

Let G be a connected and edge-colored graph of order n , where adjacent edges may be colored the same. A tree in G is a rainbow tree if all of its edges have distinct colors. Let k be an integer with 2 ≤ k ≤ n . The minimum number of colors needed in an edge coloring of G such that there exists a rainbow tree connecting S with minimum size for every k -subset S of V ( G ) is called the strong k -rainbow index of G , denoted by s r x k ( G ) . In this paper, we study the s r x 3 of edge-comb product of a path and a connected graph, denoted by P n o ⊳ e H . It is clearly that | E ( P n o ⊳ e H )| is the trivial upper bound for s r x 3 ( P n o ⊳ e H ) . Therefore, in this paper, we first characterize connected graphs H with s r x 3 ( P n o ⊳ e H )=| E ( P n o ⊳ e H )| , then provide a sharp upper bound for s r x 3 ( P n o ⊳ e H ) where s r x 3 ( P n o ⊳ e H )≠| E ( P n o ⊳ e H )| . We also provide the exact value of s r x 3 ( P n o ⊳ e H ) for some connected graphs H .