Graphs with 3-rainbow index n - 1 and n - 2

Abstract

Let G = (V (G),E(G)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) → {1, 2,. ., q}, q ∈ ℕ, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree connecting S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for each k-subset S of V (G) is called the κ-rainbow index of G, denoted by rxκ (G), where κ is an integer such that 2 ≤ κ ≤ n. Chartrand et al. got that the κ-rainbow index of a tree is n-1 and the k-rainbow index of a unicyclic graph is n-1 or n-2. So there is an intriguing problem: Characterize graphs with the κ-rainbow index n - 1 and n - 2. In this paper, we focus on κ= 3, and characterize the graphs whose 3-rainbow index is n - 1 and n - 2, respectively.