  In-phase synchronization is a stable solution of identical Kuramoto
oscillators coupled on a network with identical positive connections,
regardless of network topology. However, this fact does not mean that the
networks always synchronize in-phase because other attractors besides the
stable solution may exist. The critical connectivity $\mu_{\mathrm{c}}$ is
defined as the network connectivity above which only the in-phase solution is
stable for all the networks. In other words, below $\mu_{\mathrm{c}}$, one can
find at least one network which has a stable solution besides the in-phase
sync. The best known evaluation of the value so far is
$0.6828\cdots\leq\mu_{\mathrm{c}}\leq0.7889$. In this paper, focusing on the
twisted states of the circulant networks, we provide a method to systematically
analyze the linear stability of all possible twisted states on all possible
circulant networks. This method using integer programming enables us to find
the densest circulant network having a stable twisted state besides the
in-phase sync, which breaks a record of the lower bound of the
$\mu_{\mathrm{c}}$ from $0.6828\cdots$ to $0.6838\cdots$. We confirm the
validity of the theory by numerical simulations of the networks not converging
to the in-phase solution.
