Abstract
The acyclic chromatic number a(G) of a graph G is the minimum number of colors such that G has a proper vertex coloring and no bichromatic cycles. For a graph G with maximum degree \(\Delta \), Grünbaum (1973) conjectured \(a(G)\le \Delta +1\). Up to now, the conjecture has only been shown for \(\Delta \le 4\). In this paper, it is proved that \(a(G)\le 12\) for \(\Delta =7\), thus improving the result \(a(G)\le 17\) of Dieng et al. (in: Proc. European conference on combinatorics, graph theory and applications, 2010).
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This work was partially supported by the National Natural Science Foundation of China (nos. 11771443,12001481 and 12071265) and Shandong Province Natural Science Foundation (no. ZR2017QF011)
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Wang, J., Miao, L., Song, W. et al. Acyclic Coloring of Graphs with Maximum Degree 7. Graphs and Combinatorics 37, 455–469 (2021). https://doi.org/10.1007/s00373-020-02254-w
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DOI: https://doi.org/10.1007/s00373-020-02254-w