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Let G be a graph, m > r ,1 integers. Suppose that it has a good-coloring with m colors which uses at most r colors in the neighborhood of every vertex. We investigate these so-called local r-colorings. One of our results (Theorem 2 .4) states : The chromatic number of G, Chr(G)-r2' 1092109 2 m (and this value is the best possible in a certain sense). We… (More)

We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worst-case analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is… (More)

We address the long-standing conjecture that all permutations have polynomially bounded word length in terms of any set of generators of the symmetric group. The best available bound on the maximum required word length is exponential in <i>n</i> log <i>n</i>. Polynomial bounds on the word length have previously been established for very special classes of… (More)

We introduce new Monte Carlo methods to speed up and greatly simplify the manipulation of permutation groups. The methods are of a combinatorial character and use elementary group theory only. We achieve a nearly optimal 0(n3 loge n) running time for membership testing, an improvement of two orders of magnitude compared to known elementary algorithms and… (More)

Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online's data policy on reuse of materials please consult the policies page. Abstract. In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group… (More)

We show that the basic problems of permutation group manipulation admit efficient parallel solutions. Given a permutation group G by a list of generators, we find a set of NC-efficient strong generators in NC. Using this, we show, that the following problems are in NC: membership in G; determining the order of G; finding the center of G; finding a… (More)