  The problem of two-sender unicast index coding consists of two senders and a
set of receivers. Each receiver demands a unique message and possesses some of
the messages demanded by other receivers as its side-information. Every
demanded message is present with at least one of the senders. Senders avail the
knowledge of the side-information at the receivers to reduce the number of
broadcast transmissions. Solution to this problem consists of finding the
optimal number of coded transmissions from the two senders. One important class
of the two-sender problem consists of the messages at the senders and the
side-information at the receivers satisfying \emph{fully-participated
interactions}. This paper provides the optimal broadcast rates, for all the
unsolved cases of the two-sender problem with fully-participated interactions
when the associated \emph{interaction digraphs} contain cycles. The optimal
broadcast rates are provided in terms of those of the three independent
single-sender problems associated with the two-sender problem. This paper also
provides an achievable broadcast rate with $t$-bit messages for any finite $t$
and any two-sender problem with fully-participated interactions belonging to
$(i)$ any one of the six instances (classes) of the two-sender problem when the
associated interaction digraph does not contain any cycle, and $(ii)$ one of
the classes of the two-sender problem when the associated interaction digraph
contains cycles. The achievable broadcast rates are obtained by exploiting the
symmetries of the confusion graph to color the same according to the two-sender
graph coloring.
