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