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


Let GG be a nontrivial connected graph with an edge-coloring c:E(G)β†’{1,2,…,q},c:E(G)\rightarrow \{1,2,\ldots,q\}, q∈Nq\in \mathbb{N}, where adjacent edges may be colored the same. A tree TT in GG is a rainbowtreerainbow tree if no two edges of TT receive the same color. For a vertex set SβŠ†V(G)S\subseteq V(G), the tree connecting SS in GG is called an SS-tree. The minimum number of colors that are needed in an edge-coloring of GG such that there is a rainbow SS-tree for each kk-set SS of V(G)V(G) is called the kk-rainbow index of GG, denoted by rxk(G)rx_k(G). In \cite{Zhang}, they got that the kk-rainbow index of a tree is nβˆ’1n-1 and the kk-rainbow index of a unicyclic graph is nβˆ’1n-1 or nβˆ’2n-2. So there is an intriguing problem: Characterize graphs with the kk-rainbow index nβˆ’1n-1 and nβˆ’2n-2. In this paper, we focus on k=3k=3, and characterize the graphs whose 3-rainbow index is nβˆ’1n-1 and nβˆ’2n-2, respectively.Comment: 14 page

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