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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 $$2\le k\le n$$2≤k≤n. For a vertex subset $$S \subseteq V(G)$$S⊆V(G) with $$\left| S\right| \ge 2$$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 $$px_k(G)$$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 $$\delta \ge 3$$δ≥3, $$px_{3}(G)\le n\frac{\ln (\delta +1)}{\delta +1}(1+o_{\delta }(1))+2$$px3(G)≤nln(δ+1)δ+1(1+oδ(1))+2. We also prove that for every connected graph G with minimum degree at least 3, $$px_{3}(G) \le px_{3}(G[D])+3$$px3(G)≤px3(G[D])+3 when D is a connected 3-way dominating set of G and $$px_{3}(G) \le px_{3}(G[D])+1$$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 $$px_3(G) \le \lfloor \frac{n}{2}\rfloor $$px3(G)≤⌊n2⌋ for any 2-connected graph with at least four vertices.

Some Upper Bounds for the 3-Proper Index of Graphs