Some Upper Bounds for the 3-Proper Index of Graphs

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 px_k(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 δ≥ 3 , px_3(G)≤ nln (δ +1)/δ +1(1+o_δ(1))+2 . We also prove that for every connected graph G with minimum degree at least 3, px_3(G) ≤ px_3(G[D])+3 when D is a connected 3-way dominating set of G and px_3(G) ≤ px_3(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) ≤⌊n/2⌋ for any 2-connected graph with at least four vertices.