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).

中文翻译：

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最大度数为7的图的非循环着色

图G的非循环色数a（G）是最小颜色数，使得G具有适当的顶点着色并且没有双色循环。对于具有最大度数\（\ Delta \）的图G，Grünbaum（1973）猜想\（a（G）\ le \ Delta +1 \）。到目前为止，只显示了\（\ Delta \ le 4 \）的猜想。本文证明了\（\ Delta = 7 \）的\（a（G）\ le 12 \ ），从而提高了Dieng等人的结果\（a（G）\ le 17 \）。（在：Proc。欧洲组合论，图论和应用会议，2010年）。