
Almost EnvyFree Allocations with Connected Bundles
We study the existence of allocations of indivisible goods that are envy...
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Cycle Extendability of Hamiltonian Strongly Chordal Graphs
In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycl...
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The Minimality of the GeorgesKelmans Graph
In 1971, Tutte wrote in an article that "it is tempting to conjecture th...
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Hamiltonian chromatic number of block graphs
Let G be a simple connected graph of order n. A hamiltonian coloring c o...
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Reachability and Coverage Planning for Connected Agents: Extended Version
Motivated by the increasing appeal of robots in informationgathering mi...
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Disproportionate division
We study the disproportionate version of the classical cakecutting prob...
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Envyfree division of multilayered cakes
We study the problem of dividing a multilayered cake among heterogeneou...
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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 envyfree 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, many of which are graphs. This correspondence links EF fair division of tangles to EFk_outer fair division of graphs. We know from Bilò 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 nonstringable 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 nonstringable topological class 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, and explains why a focus on Hamiltonian graphs was necessary. 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 nonstringable lips tangle guarantees envyfree allocations of connected shares for three agents. We then modify the discrete version of Stromquist's procedure in Bilò et al to show that all graphs in the topological class guarantee EF1_outer allocations for three agents.
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