Fair division of graphs and of tangled cakes

Abstract

A tangle is a connected topological space constructed by gluing together several copies of the unit interval [0, 1]. We explore which tangles guarantee envy-free (aka EF) 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 T corresponds in a natural way to an infinite topological class G(T ) of multigraphs, infinitely many of which are graphs. This correspondence links EF fair division of tangles to EFk outer fair division of graphs: the vertices of a graph, treated as indivisible objects, are allocated to the agents, each agent's share of vertices is contiguous (connected as an induced subgraph), no agent envies another's share after she pretends some selection of k or fewer vertices disappear from that share, and that disappearance does not destroy contiguity. We know from Bilò et al [3] that all Hamiltonian graphs guarantee EF1 outer allocations when the number of agents is 2, 3, or 4 and guarantee EF2 outer allocations for arbitrarily many agents.We show that exactly six tangles are stringable; these guarantee EF allocations of connected shares for any number of agents, and their associated topological classes contain only Hamiltonian graphs. Any non-stringable tangle T 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 G(T ) are not Hamiltonian, and a negative transfer theorem shows that for each k ≥ 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 Bilò et al [3], and explains why a focus on Hamiltonian graphs was necessary for certain results in that paper.With bounds on the number of agents, however, we obtain positive results for some nonstringable classes. An elaboration of Stromquist's moving knife procedure shows that the non-stringable lips tangle L guarantees envy-free allocations of connected shares for three agents. We then modify the discrete version of Stromquist's procedure in Bilò et al [3] to show that all graphs in the topological class G(L) (most of which are non-Hamiltonian) guarantee EF1 outer allocations for three agents.