The vertex-arboricity a ( G ) of a graph G is the minimum number of colors required for a vertex coloring of G such that no cycle is monochromatic. The list vertex-arboricity a l ( G ) is the list-coloring version of this concept. In this paper, we prove that every planar graph G without intersecting 5-cycles has a l ( G ) ≤ 2. This extends a result by Raspaud and Wang [On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008), 1064-1075] that every planar graph G without 5-cycles has a(G) ≤ 2.

中文翻译：

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列出不相交 5 圈的平面图的顶点树性

图 G 的顶点树状 a ( G ) 是 G 的顶点着色所需的最小颜色数，使得没有循环是单色的。list vertex-arboricity al ( G ) 是这个概念的列表着色版本。在本文中，我们证明了每个不与 5 圈相交的平面图 G 都有 al ( G ) ≤ 2。这扩展了 Raspaud 和 Wang [On the vertex-arboricity of plane graphs, European J. Combin. 29 (2008), 1064-1075] 每个没有 5 圈的平面图 G 都有 a(G) ≤ 2。