A tangle is a connected topological space constructed by gluing several copies of the unit interval $[0, 1]$. We explore which tangles guarantee envy-free allocations of connected shares for n agents, meaning that such allocations exist no matter which monotonic and continuous functions represent agents' valuations. Each single tangle $\mathcal{T}$ corresponds in a natural way to an infinite topological class $\mathcal{G}(\mathcal{T})$ of multigraphs, many of which are graphs. This correspondence links EF fair division of tangles to EFk$_{outer}$ fair division of graphs. We know from Bil\`o et al that all Hamiltonian graphs guarantee EF1$_{outer}$ allocations when the number of agents is 2, 3, 4 and guarantee EF2$_{outer}$ allocations for arbitrarily many agents. We show that exactly six tangles are stringable; these guarantee EF connected allocations for any number of agents, and their associated topological classes contain only Hamiltonian graphs. Any non-stringable tangle has a finite upper bound r on the number of agents for which EF allocations of connected shares are guaranteed. Most graphs in the associated non-stringable topological class are not Hamiltonian, and a negative transfer theorem shows that for each $k \geq 1$ most of these graphs fail to guarantee EFk$_{outer}$ allocations of vertices for r + 1 or more agents. This answers a question posed in Bil\`o et al, and explains why a focus on Hamiltonian graphs was necessary. With bounds on the number of agents, however, we obtain positive results for some non-stringable classes. An elaboration of Stromquist's moving knife procedure shows that the non-stringable lips tangle guarantees envy-free allocations of connected shares for three agents. We then modify the discrete version of Stromquist's procedure in Bil\`o et al to show that all graphs in the topological class guarantee EF1$_{outer}$ allocations for three agents.

中文翻译：

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图形和纠结蛋糕的公平划分

缠结是通过胶合单位间隔$ [0，1] $的多个副本而构成的连通拓扑空间。我们探讨了哪个缠结确保n个代理无关联分配的羡慕分配，这意味着无论哪个单调函数和连续函数代表代理的估值，这种分配都存在。每个缠结$ \ mathcal {T} $自然地对应于多图的无限拓扑类$ \ mathcal {G}（\ mathcal {T}）$，其中许多是图。该对应关系将纠缠的EF公平分割与图的EFk $ _ {outer} $公平分割联系起来。从Bil \`o等人知道，当代理人数为2、3、4时，所有汉密尔顿图都保证EF1 $ _ {outer} $分配，并保证任意多个代理的EF2 $ _ {{outer} $]分配。我们证明恰好有六个缠结是可串的。这些保证了EF连接可以分配给任何数量的代理，并且它们的关联拓扑类别仅包含哈密顿图。任何不可串缠的缠结在代理数量上都有一个有限的上限r，可以为之保证已连接份额的EF分配。关联的非字符串化拓扑类别中的大多数图不是哈密顿量，并且负转移定理表明，对于每个$ k \ geq 1 $，大多数这些图都不能保证r + 1的顶点EFk $ _ {outer} $个分配或更多代理商。这回答了Bilo等人提出的问题，并解释了为什么必须要关注哈密顿图。但是，随着代理数量的限制，对于某些不可字符串化的类，我们获得了积极的结果。斯特龙奎斯特的详细说明 动刀过程表明，不可拉紧的双唇缠结可确保为三个代理分配关联股的无羡慕的分配。然后，我们在Bil \ o等人中修改了Stromquist过程的离散版本，以表明拓扑类中的所有图都保证了三个代理的EF1 $ _ {outer} $分配。