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

Abstract

Let G be a nontrivial connected graph of order n. Let k be an integer with 2 ≤ k ≤ n. A strong k-rainbow coloring of G is an edge-coloring of G having property that for every set S of k vertices of G, there exists a tree with minimum size containing S whose all edges have distinct colors. The minimum number of colors required such that G admits a strong k-rainbow coloring is called the strong k-rainbow index srxk (G) of G. In this paper, we study the strong 3-rainbow index of comb product between a tree and a connected graph, denoted by Tn ⊲o H. Notice that the size of Tn ⊲o H is the trivial upper bound for srx3 (Tn ⊲o H), which means we can assign distinct colors to all edges of Tn ⊲o H. However, there are some connected graphs H such that some edges of Tn ⊲o H may be colored the same. Therefore, in this paper, we characterize connected graphs H with srx3 (Tn ⊲o H) = |E(Tn ⊲o H)|. We also provide a sharp upper bound for srx3 (Tn ⊲o H) where srx3 (Tn ⊲o H) ≠ |E(Tn ⊲o H)|. In addition, we determine the srx3 (Tn ⊲o H) for some connected graphs H.