  We study the existence of allocations of indivisible goods that are envy-free
up to one good (EF1), under the additional constraint that each bundle needs to
be connected in an underlying item graph. If the graph is a path and the
utility functions are monotonic over bundles, we show the existence of EF1
allocations for at most four agents, and the existence of EF2 allocations for
any number of agents; our proofs involve discrete analogues of the Stromquist's
moving-knife protocol and the Su--Simmons argument based on Sperner's lemma.
For identical utilities, we provide a polynomial-time algorithm that computes
an EF1 allocation for any number of agents. For the case of two agents, we
characterize the class of graphs that guarantee the existence of EF1
allocations as those whose biconnected components are arranged in a path; this
property can be checked in linear time.
