Graphs with 3-rainbow index n-1 and n-2
Xue Liang Li, Ingo Schiermeyer, Kang Yang, Yan Zhao · Discussiones Mathematicae Graph Theory · 2014
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 k-rainbow index of G, denoted by rx k (G), where k is an integer such that 2 k n. Chartrand et al. got that the k-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 k-rainbow index n -1 and n -2. In this paper, we focus on k = 3, and characterize the graphs whose 3-rainbow index is n -1 and n -2, respectively.