Fair division of graphs and of tangled cakes

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.