Abstract
An r-dynamick-coloring of a graphG is a proper k-coloring such that every vertex v in V(G) has neighbors in at least \(min\{d(v),r\}\) different classes. The r-dynamic chromatic number ofG, written \(\chi _{r}(G)\), is the minimum integer k such that G has such a coloring. In this paper, we investigate the r-dynamic \((r+1)\)-coloring (i.e. optimal r-dynamic coloring) of sparse graphs and prove that \(\chi _{r}(G)\le r+1\) holds if G is a planar graph with \(g(G)\ge 7\) and \(r\ge 16\), which is a generalization of the case \(r=\Delta \).
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Acknowledgements
This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant Nos. LQ19A010005, LY17A010025, General SRT Program of Jiaxing University in 2018, and National Science Foundation of China under Grant No. 11771403.
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Yi, D., Zhu, J., Feng, L. et al. Optimal r-dynamic coloring of sparse graphs. J Comb Optim 38, 545–555 (2019). https://doi.org/10.1007/s10878-019-00387-0
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DOI: https://doi.org/10.1007/s10878-019-00387-0