Abstract
Let k and r be two positive integers. An r-dynamic coloring of a graph G is a proper k-coloring \(\varphi \) such that \(\mid \varphi (N_{G}(v))\mid \ge \) min\(\{d_{G}(v),r\}\) for each \(v\in V(G)\). In this paper, we study the r-dynamic coloring of planar graphs without 3-,5-cycle, and 4-cycle is not adjacent to \(7^{-}\)-cycles. We prove that the upper bound of r-dynamic chromatic number of such graph is \(r+3\) if \(r\ge 14\).
This work is supported by a research grant NSFC (11271334).
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Bu, Y., Yang, R., Zhu, H. (2022). The r-Dynamic Chromatic Number of Planar Graphs Without Special Short Cycles. In: Ni, Q., Wu, W. (eds) Algorithmic Aspects in Information and Management. AAIM 2022. Lecture Notes in Computer Science, vol 13513. Springer, Cham. https://doi.org/10.1007/978-3-031-16081-3_27
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DOI: https://doi.org/10.1007/978-3-031-16081-3_27
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