About randomised distributed graph colouring and graph partition algorithms

We present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time O ( log n ) with probability 1 - o ( n - 1 ) . Each message containing 1 bit, its bit complexity per channel is O ( log n ) . From this algorithm, we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree Δ and size n that uses at most Δ + 1 colours and halts in time O ( log n ) with probability 1 - o ( n - 1 ) . We also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time O ( log n ) with probability 1 - o ( n - 1 ) . We study some parameters such as the number, the size and the radius of trees of the spanning forest.