Abstract
We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree T of order n, \( cfc (T)\ge cfc (P_n)=\lceil \log _{2} n\rceil \), which completely confirms the conjecture of Li and Wu [24]. We then present a sharp upper bound for the conflict-free connection number of trees by a simple algorithm. Furthermore, we show that the conflict-free connection number of the binomial tree with \(2^{k-1}\) vertices is \(k-1\). At last, we study trees which are \( cfc \)-critical, and prove that if a tree T is \( cfc \)-critical, then the conflict-free connection coloring of T is equivalent to the edge ranking of T.
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The authors would like to thank the reviewers for their valuable suggestions and comments, which helped to improve the presentation of the paper.
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Supported by NSFC Nos. 11871034, 11531011 and NSFQH No. 2017-ZJ-790.
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Chang, H., Ji, M., Li, X. et al. Conflict-free connection of trees. J Comb Optim 42, 340–353 (2021). https://doi.org/10.1007/s10878-018-0363-x
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DOI: https://doi.org/10.1007/s10878-018-0363-x