  In the dispersion problem, a set of $k$ co-located mobile robots must
relocate themselves in distinct nodes of an unknown network. The network is
modeled as an anonymous graph $G=(V,E)$, where the nodes of the graph are not
labeled. The edges incident to a node $v$ with degree $d$ are labeled with port
numbers in the range $0,1, \cdots, d-1$ at $v$. The robots have unique ids in
the range $[0,L]$, where $L \ge k$, and are initially placed at a source node
$s$. Each robot knows only its own id but does not know the ids of the other
robots or the values of $L,k$. The task of dispersion was traditionally
achieved with the assumption of two types of communication abilities: (a) when
some robots are at the same node, they can communicate by exchanging messages
between them (b) any two robots in the network can exchange messages between
them.
  In this paper, we ask whether this ability of communication among co-located
robots is necessary to achieve dispersion. We show that even if the ability of
communication is not available, the task of dispersion by a set of mobile
robots can be achieved in a much weaker model where a robot at a node $v$ has
the access of following very restricted information at the beginning of any
round: (1) am I alone at $v$? (2) the number of robots at $v$ increased or
decreased compare to the previous round?
  We propose a deterministic algorithm that achieves dispersion on any given
graph $G=(V,E)$ in time $O\left( k\log L+k^2 \log \Delta\right)$, where
$\Delta$ is the maximum degree of a node in $G$. Each robot uses $O(\log L+
\log \Delta)$ additional memory. We also prove that the task of dispersion
cannot be achieved by a set of mobile robots with $o(\log L + \log \Delta)$
additional memory.
