  We prove three new lower bounds for graph connectivity in the $1$-bit
broadcast congested clique model, BCC$(1)$. First, in the KT-$0$ version of
BCC$(1)$, in which nodes are aware of neighbors only through port numbers, we
show an $\Omega(\log n)$ round lower bound for CONNECTIVITY even for
constant-error randomized Monte Carlo algorithms. The deterministic version of
this result can be obtained via the well-known "edge-crossing" argument, but,
the randomized version of this result requires establishing new combinatorial
results regarding the indistinguishability graph induced by inputs. In our
second result, we show that the $\Omega(\log n)$ lower bound result extends to
the KT-$1$ version of the BCC$(1)$ model, in which nodes are aware of IDs of
all neighbors, though our proof works only for deterministic algorithms. Since
nodes know IDs of their neighbors in the KT-$1$ model, it is no longer possible
to play "edge-crossing" tricks; instead we present a reduction from the 2-party
communication complexity problem PARTITION in which Alice and Bob are give two
set partitions on $[n]$ and are required to determine if the join of these two
set partitions equals the trivial one-part set partition. While our KT-$1$
CONNECTIVITY lower bound holds only for deterministic algorithms, in our third
result we extend this $\Omega(\log n)$ KT-1 lower bound to constant-error Monte
Carlo algorithms for the closely related CONNECTED COMPONENTS problem. We use
information-theoretic techniques to obtain this result. All our results hold
for the seemingly easy special case of CONNECTIVITY in which an algorithm has
to distinguish an instance with one cycle from an instance with multiple
cycles. Our results showcase three rather different lower bound techniques and
lay the groundwork for further improvements in lower bounds for CONNECTIVITY in
the BCC$(1)$ model.
