  We study fair allocation of indivisible items, where the items are furnished
with a set of conflicts, and agents are not permitted to receive conflicting
items. This kind of constraint captures, for example, participating in events
that overlap in time, or taking on roles in the presence of conflicting
interests. We demonstrate, both theoretically and experimentally, that fairness
characterizations such as EF1, MMS and MNW still are applicable and useful
under item conflicts. Among other existence, non-existence and computability
results, we show that a $1/\Delta$-approximate MMS allocation for $n$ agents
may be found in polynomial time when $n>\Delta>2$, for any conflict graph with
maximum degree $\Delta$, and that, if $n > \Delta$, a 1/3-approximate MMS
allocation always exists.
