Abstract In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. The conjecture is still wide open, and, as a possible approach to it, many statements that are equivalent or related to it have been studied. In this paper, we extend the statement to the class of line graphs of 3-hypergraphs, and generalize it to Tutte cycles and paths (note that a line graph of a 3-hypergraph is K 1 , 4 -free but can contain induced claws K 1 , 3 , and that a Tutte cycle/path is a cycle/path such that any component of its complement has at most three vertices of attachment). Among others, we formulate the following conjectures: (i) every 2-connected line graph of a 3-hypergraph has a Tutte maximal cycle containing any two prescribed vertices, (ii) every 3-connected line graph of a 3-hypergraph has a Tutte maximal cycle containing any three prescribed vertices, (iii) every connected line graph of a 3-hypergraph has a Tutte maximal ( a , b ) -path for any two vertices a , b , (iv) every 4-connected line graph of a 3-hypergraph is Hamilton-connected, and we show that all these (seemingly much stronger) statements are equivalent with Thomassen’s conjecture.

中文翻译：

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3-超图线图的 Thomassen 猜想

摘要 1986 年，Thomassen 猜想每个 4-连通线图都是哈密顿图。这个猜想仍然是开放的，作为一种可能的方法，已经研究了许多与它等效或相关的陈述。在本文中，我们将陈述扩展到 3-超图的线图类，并将其推广到 Tutte 循环和路径（注意，3-超图的线图是 K 1 , 4 -free 但可以包含诱导爪K 1 , 3 ，并且 Tutte 循环/路径是这样的循环/路径，使得其补集的任何组件最多具有三个连接顶点）。其中，我们提出以下猜想：(i) 3-hypergraph 的每个 2-connected 线图都有一个 Tutte 最大循环，其中包含任意两个规定的顶点，