Abstract
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 \(2\le k\le n\). For a vertex subset \(S \subseteq V(G)\) with \(\left| S\right| \ge 2\), 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 \(\delta \ge 3\), \(px_{3}(G)\le n\frac{\ln (\delta +1)}{\delta +1}(1+o_{\delta }(1))+2\). We also prove that for every connected graph G with minimum degree at least 3, \(px_{3}(G) \le px_{3}(G[D])+3\) when D is a connected 3-way dominating set of G and \(px_{3}(G) \le px_{3}(G[D])+1\) 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 \(px_3(G) \le \lfloor \frac{n}{2}\rfloor \) for any 2-connected graph with at least four vertices.
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The authors would like to thank the reviewers for their helpful suggestions and comments.
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Communicated by Sandi Klavžar.
This work was supported by NSFC Nos. 11371205 and 11531011, “973” Program No. 2013CB834204, and PCSIRT.
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Chang, H., Li, X. & Qin, Z. Some Upper Bounds for the 3-Proper Index of Graphs. Bull. Malays. Math. Sci. Soc. 41, 1681–1695 (2018). https://doi.org/10.1007/s40840-016-0404-5
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DOI: https://doi.org/10.1007/s40840-016-0404-5