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The k-proper index of graphs

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  • Chen, Lin
  • Li, Xueliang
  • Liu, Jinfeng

Abstract

A tree T in an edge-colored graph is a proper tree if any two adjacent edges of T are colored with different colors. Let G be a graph of order n and k be a fixed integer with 2 ≤ k ≤ n. For a vertex set S ⊆ V(G), a tree containing 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 set S of k vertices in G, there exists a proper S-tree in G. The k-proper index of a nontrivial connected graph G, denoted by pxk(G), is the smallest number of colors needed in a k-proper coloring of G. In this paper, we state some simple observations about pxk(G) for a nontrivial connected graph G. Meanwhile, the k-proper indices of some special graphs are determined, and for every pair of positive integers a, b with 2 ≤ a ≤ b, a connected graph G with pxk(G)=a and rxk(G)=b is constructed for each integer k with 3 ≤ k ≤ n. Also, we characterize the graphs with k-proper index n−1 and n−2, respectively.

Suggested Citation

  • Chen, Lin & Li, Xueliang & Liu, Jinfeng, 2017. "The k-proper index of graphs," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 57-63.
    • Handle: RePEc:eee:apmaco:v:296:y:2017:i:c:p:57-63
    DOI: 10.1016/j.amc.2016.10.025
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    References listed on IDEAS

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    1. Li, Shasha & Li, Xueliang & Shi, Yongtang, 2015. "Note on the complexity of deciding the rainbow (vertex-) connectedness for bipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 155-161.
    2. Qingqiong Cai & Xueliang Li & Yan Zhao, 2016. "The 3-rainbow index and connected dominating sets," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1142-1159, April.
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    Cited by:

    1. Hong Chang & Xueliang Li & Colton Magnant & Zhongmei Qin, 2018. "The $$(k,\ell )$$ ( k , ℓ ) -proper index of graphs," Journal of Combinatorial Optimization, Springer, vol. 36(2), pages 458-471, August.

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