Coloring permutation graphs in parallel

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having speci/c key properties. We propose an e0cient parallel algorithm which colors an n-node permutation graph in O(log n) time using O(n=log n) processors on the CREW PRAM model. Speci/cally, given a permutation we construct a tree T∗[ ], which we call coloring-permutation tree, using certain combinatorial properties of . We show that the problem of coloring a permutation graph is equivalent to /nding vertex levels in the coloring-permutation tree. ? 2002 Elsevier Science B.V. All rights reserved.