A note on the 3-rainbow index of complete bipartite graphs

摘要

A tree T, in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. For a vertex subset S ∈ V(G), a tree that connects 5 in G is called an S-tree. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow S-tree T in G. The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rx_k(G). It is NP-hard to compute the rx_k(G) for the general graphs G. We consider the 3-rainbow index of complete bipartite graphs K_(s,t). For 3 ≤ s ≤ t, we have determined the tight bounds of rx_3(K_(s,t))- In this paper, we continue the study. For 2 = s ≤ t, we develop a converse idea and apply with the model of chessboard to study the problem. Finally, we obtain the exact value of rx_3(K_(s,t)) with 2 = s ≤ t.