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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 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)
The diameter of a group G with respect to a set S of generators is the maximum over g E G of the length of the shortest word in S U S-' representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. " Best " generators (giving minimum diameter while keeping the number of generators limited) are(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)
Given a black-box group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm. The running time is polynomial in the input length and in the time requirement for the group operations in G. The algorithm chooses a relatively small number of (nearly) uniformly distributed(More)