Jiang [The synchronization of non uniform networks of finite automata, in: FOCS, Vol. 89, 1989, pp. 376–381, The synchronization of non uniform networks of finite automata, Inform. and Comput. 97(2) (1992) 234–261] proved a remarkable result: for every k , there exists a cellular automaton synchronizing every degree ⩽ k connected graph with arbitrary symmetric communication delays. The synchronization time obtained by Jiang is O ( Δ 3 ) where Δ is the maximum communication delay between two cells. Mazoyer [Synchronization of a line of finite automata with non uniform delays, 1990, unpublished] proved an O ( D 2 ) synchronization time where D is the sum of the communication delays of the degree ⩽ k connected graph (together with an O ( D log D ) synchronization time in case the graph has only two cells). In this paper, we prove (cf. Theorem 23) that for any m ⩾ 2 one can synchronize in time D ⌊ log m ( D ) ⌋ all lines of total communication delay > m 9 (shorter lines being synchronized in time 4 D ). A result which extends to bounded degree connected graphs using Rosensthiel's technique [P. Rosenstiehl, Existence d’automates capables de s’accorder bien qu’arbitrairement connectés et nombreux, Internat. Comput. Center Bull. 5 (1966) 245–261, P. Rosenstiehl, J.R. Fiksel, A. Holliger, Intelligent graphs: networks of finite automata capable of solving graph problems, in: R.C. Read (Ed.), Graph Theory and Computing, Academic Press, New York, 1972, pp. 219–265]. As shown by Vivien [Cellular Automata: A Geometrical Approach, to appear], this result is already optimal for lines of two cells with arbitrary communication delay. The method relies heavily on Jiang technique of circuit with revolving information.