Given a graph G and a list assignment L(v) for each vertex of v of G, a proper L-list-coloring of G is a function that maps every vertex to a color in L(v) such that no pair of adjacent vertices have the same color. We say that a graph is k-list-colorable when every vertex v has a list of colors of size at least k. A 2-distance coloring is a coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance list (\(\Delta +2\))-coloring for planar graphs with girth at least 10 and maximum degree \(\Delta \ge 4\).

中文翻译：

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2-距离列表 $$(\Delta +2)$$ ( Δ + 2 ) - 周长至少为 10 的平面图的着色

给定图G和G的v的每个顶点的列表分配L ( v ) ，G的适当L -list-coloring是将每个顶点映射到L ( v ) 中的颜色的函数，使得没有一对相邻的顶点具有相同的颜色。当每个顶点v都有一个大小至少为k的颜色列表时，我们说一个图是k -list-colorable的。2 距离着色是距离最多为 2 的顶点不能共享相同颜色的着色。我们证明了一个 2 距离列表 ( \(\Delta +2\))-对周长至少为 10 且最大度数为\(\Delta \ge 4\)的平面图着色。