Before revision:  to the best of our knowledge, the state-of-the-art distributed algorithm for this problem with a message complexity of o  (m^{3/4}) incurs a round complexity of \Omega(m^{1/3 (m^{2/3) and an amortized round complexityof } under fully dynamic edge updates in the CONGEST model, where (\cdot) suppresses polylogarithmic factors. 
After revision:  to the best of our knowledge, the state-of-the-art distributed algorithm for this problem with a message complexity of o (m^{3/4}) incurs a round complexity of \Omega(m^{1/3 (m^{2/3) and an amortized round complexityof } under fully dynamic edge updates in the CONGEST model, where amortized messages in O(1) amortized rounds, i.e., we achieve a polynomial improvement in the message complexity by a polylog n increase to the round complexity; moreover, the algorithm of Assadi et al. makes an implicit assumption that the