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Conflict-free Connection Number and Independence Number of a Graph

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Abstract

An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that cfc(G) ≤ α(G) for any connected graph G, and give an example to show that the bound is sharp. With this result, we prove that if T is a tree with \(\Delta(T)\geq\frac{\alpha(T)+2}{2}\), then cfc(T) = Δ(T).

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Acknowledgments

This work was done during the first author is visiting Nankai University. The authors would like to express their sincere thanks to Professor Xueliang Li for his helpful suggestions, and to the anonymous referees for their helpful comments to improve the paper.

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Authors and Affiliations

  1. School of Computer Engineering and Applied Mathematics, Changsha University, Changsha, 410022, China

    Jing Wang

  2. Hunan Province Key Laboratory of Industrial Internet Technology and Security, Changsha University, Changsha, 410022, China

    Jing Wang

  3. College of Mathematical Science, Tianjin Normal University, Tianjin, 300007, China

    Meng Ji

Authors
  1. Jing Wang
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  2. Meng Ji
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Correspondence to Jing Wang.

Additional information

This paper is supported by Hunan Education Department Foundation (No. 18A382).

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Wang, J., Ji, M. Conflict-free Connection Number and Independence Number of a Graph. Acta Math. Appl. Sin. Engl. Ser. 37, 278–286 (2021). https://doi.org/10.1007/s10255-021-1013-0

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  • DOI: https://doi.org/10.1007/s10255-021-1013-0

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