A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at distance at most 2 from each other receive different colors. Suppose that G is a planar graph with girth 5 and maximum degree \(\Delta \). We prove that G admits a 2-distance \(\Delta +7\) coloring, which improves the result of Dong and Lin (J Comb Optim 32(2):645–655, 2016). Moreover, we prove that G admits a 2-distance \(\Delta +6\) coloring when \(\Delta \ge 10\).

中文翻译：

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周长为 5 的平面图的 2 距离着色的一些结果

如果图G的任意两个距离最多为 2 的顶点接收到不同的颜色，则图 G的顶点着色称为 2 距离着色。假设G是周长为 5 且最大度为\(\Delta \)的平面图。我们证明G允许 2 距离\(\Delta +7\)着色，这改进了 Dong 和 Lin 的结果 (J Comb Optim 32(2):645–655, 2016)。此外，我们证明当\(\Delta \ge 10\)时， G允许 2 距离\(\Delta +6\)着色。