Abstract Let G be a connected simple graph with at least one edge. The hypergraph H = H ( G ) with the same vertex set as G whose hyper-edges are the maximal cliques of G is called the clique-hypergraph of G . A list-assignment of G is a function L which assigns to each vertex v ∈ V ( G ) a set L ( v ) (called the list of v ). A k -list-assignment of G is a list-assignment L such that L ( v ) has at least k elements for every v ∈ V ( G ) . For a given list assignment L , a list-coloring of H ( G ) is a function c : V ( G ) → ∪ v L ( v ) such that c ( v ) ∈ L ( v ) for every v ∈ V ( G ) and no hyper-edge of H ( G ) is monochromatic. A list-coloring of H ( G ) is strong if no 3-cycle of G is monochromatic. H ( G ) is (strongly) k -choosable if, for every k -list assignment L , there exists a (strong) list-coloring of H ( G ) . Mohar and S ˇ krekovski proved that the clique-hypergraphs of planar graphs are strongly 4-choosable (Electr. J. Combin. 6 (1999), #R26). In this paper we give a short proof of the result and present a linear time algorithm for the strong list-4-coloring of H ( G ) if G is a planar graph. In addition, we prove that H ( G ) is strongly 4-choosable if G is a K 5 -minor-free graph, which is a generalization of their result.

中文翻译：

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强烈列出 K5-minor-free 图的着色团超图

摘要 设 G 是一个至少有一条边的连通简单图。超图 H = H ( G ) 与 G 的顶点集相同，其超边是 G 的最大团，称为 G 的团超图。G 的列表分配是一个函数 L，它为每个顶点 v ∈ V ( G ) 分配一个集合 L ( v )（称为 v 的列表）。G 的 k -list-assignment 是一个 list-assignment L，使得 L ( v ) 对于每个 v ∈ V ( G ) 至少有 k 个元素。对于给定的列表分配 L ，H ( G ) 的列表着色是一个函数 c : V ( G ) → ∪ v L ( v ) 使得 c ( v ) ∈ L ( v ) 对于每个 v ∈ V ( G ) 并且没有 H ( G ) 的超边是单色的。如果 G 的 3 个循环都不是单色的，则 H ( G ) 的列表着色很强。如果对于每个 k 列表分配 L ，存在 H ( G ) 的（强）列表着色，则 H ( G ) 是（强）k 可选的。Mohar 和 Sˇ krekovski 证明了平面图的团超图是强 4-choosable 的（Electr. J. Combin. 6 (1999), #R26）。在本文中，我们给出了结果的简短证明，并提出了一种线性时间算法，用于对 H ( G ) 进行强列表 4 着色，如果 G 是平面图。此外，我们证明了 H ( G ) 是强 4-choosable 如果 G 是 K 5 -minor-free 图，这是他们结果的推广。