Some Upper Bounds for the 3-Proper Index of Graphs

A tree T in an edge-colored graph is a proper tree if no two adjacent edges of T receive the same color. Let G be a connected graph of order n and k be a fixed integer with 2 = k = n. For a vertex subset S. V(G) with | S| = 2, a tree containing all the vertices of S in G is called an S-tree. An edge-coloring of G is called a k-proper coloring if for every k-subset S of V(G), there exists a proper S-tree in G. For a connected graph G, the k-proper index of G, denoted by pxk (G), is the smallest number of colors that are needed in a k-proper coloring of G. In this paper, we show that for every connected graph G of order n and minimum degree d = 3, px3(G) = n ln(d+ 1) d+ 1 (1 + od(1)) + 2. We also prove that for every connected graph G with minimum degree at least 3, px3(G) = px3(G[ D]) + 3 when D is a connected 3-way dominating set of G and px3(G) = px3(G[ D]) + 1 when D is a connected 3-dominating set of G. In addition, we obtain sharp upper bounds of the 3-proper index for two special graph classes: threshold graphs and chain graphs. Finally, we prove that px3(G) = n 2 for any 2-connected graph with at least four vertices.