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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 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)

Let w be a non-trivial word in two variables. We prove that the probability that two randomly chosen elements x, y of a nonabelian finite simple group S satisfy w(x, y) = 1 tends to 0 as |S| → ∞. As a consequence, we obtain a new short proof of a well-known conjecture of Magnus concerning free groups, as well as some applications to profinite groups.

Ž. We associate a weighted graph ⌬ G to each finite simple group G of Lie type. Ž. We show that, with an explicit list of exceptions, ⌬ G determines G up to Ž. isomorphism, and for these exceptions, ⌬ G nevertheless determines the characteristic of G. This result was motivated by algorithmic considerations. We prove that for any finite simple group G of Lie… (More)

We give a presentation of length O log 2 |G| for the groups G ∼ = PSU 3 (q). This result has applications in recent algorithms to compute the structure of permutation groups and matrix groups.