Abstract An R E - m -coloring of a graph G is a vertex m -coloring of G , which is relaxed (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). In this article, we prove that every planar graph with minimum degree at least 2 and girth at least 8 has an R E - m -coloring for each integer m ≥ 4 . We use the discharging method and Hall’s Theorem to simply the structures of counterexamples.

中文翻译：

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周长至少为 8 的平面图的宽松公平着色

摘要 图 G 的 RE - m -着色是 G 的顶点 m -着色，它是松弛的（每个顶点与最多一个邻居共享相同的颜色）且公平（所有颜色类别的大小最多相差一个） ）。在本文中，我们证明了每个最小度数至少为 2 且周长至少为 8 的平面图对于每个整数 m ≥ 4 都有一个 RE-m 着色。我们使用放电法和霍尔定理来简化反例的结构。