Colouring perfect planar graphs in parallel
Our algorithm assumes that the perfect planar graph is presented in the form of a planar drawing. However, this is not strictly necessary, for by  the problem of constructing a planar drawing of a planar graph is in NC; in particular, this problem can be solved in O(log2 n) time using O(n) processors on a CREW PRAM.
The original version of this algorithm bypassed the construction of the randomized algorithm, by using a different method of selecting a "nice" set of vertices at which to simultaneously perform α–β interchanges, as in the phase of interchanges in Stage 3. Here, a planar multigraph was constructed with vertices corresponding to the α–β components, with edges joining two vertices if and only if the corresponding α–β components had a common adjacent uncoloured vertex. This multigraph was then coloured using at most 5 colours and our "nice" set of vertices was chosen using this colouring. For more details, see . We are indebted to an anonymous referee for pointing out Luby's technique and thus speeding up the algorithm considerably.
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