The k-proper index of complete bipartite and complete multipartite graphs

Let G be an edge-colored graph. A tree T in G is a proper tree if no two adjacent edges of it are assigned the same color. Let k be a fixed integer with 2 ≤ k ≤ n. For a vertex subset S ⊆ V (G) with |S| ≥ 2, a tree is called an S-tree if it connects the vertices of S in G. A k-proper 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 proper S-tree T in G. The minimum number of colors that are required in a k-proper coloring of G is defined as the k-proper index of G, denoted by pxk(G). In this paper, we determine the 3-proper index of all complete bipartite and complete multipartite graphs and partially determine the k-proper index of them for k ≥ 4.