Abstract
A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|, r} colors. The r-acyclic chromatic number of G is the least number of colors in an r-acyclic coloring of G. We prove that for any number r ≥ 4, the r-acyclic chromatic number of any graph G with maximum degree Δ ≥ 7 and with girth at least (r − 1)Δ is at most (4r − 3)Δ.
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This work is supported by NSFC (Grant number 11571258) and SDNSF (Grant number ZR2016AM01).
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Cai, Js., Zhu, Xd. Improved Upper Bound for Generalized Acyclic Chromatic Number of Graphs. Acta Math. Appl. Sin. Engl. Ser. 34, 798–800 (2018). https://doi.org/10.1007/s10255-018-0791-5
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DOI: https://doi.org/10.1007/s10255-018-0791-5