Graphs with 4-rainbow index 3 and n-1

Abstract

Let G be a nontrivial connected graph with an edge-coloring c : E(G)→ {1, 2, . . . , q}, q ε N, where adjacent edges may be colored the same. A tree T in G is called a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree that connects 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 every set S of k vertices of V (G) is called the k-rainbow index of G, denoted by rxk(G). Notice that a lower bound and an upper bound of the k-rainbow index of a graph with order n is k-1 and n-1, respectively. Chartrand et al. got that the k-rainbow index of a tree with order n is n-1 and the k-rainbow index of a unicyclic graph with order n is n-1 or n-2. Li and Sun raised the open problem of characterizing the graphs of order n with rxk(G) = n-1 for k ≥ 3. In early papers we characterized the graphs of order n with 3-rainbow index 2 and n-1. In this paper, we focus on k = 4, and characterize the graphs of order n with 4-rainbow index 3 and n-1, respectively.