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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)
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)
A base of a permutation group G is a subset B of the permutation domain such that only the identity of G fixes B pointwise. The permutation representations of important classes of groups, including all finite simple groups other than the alternating groups, admit O(log n) size bases, where n is the size of the permutation domain. Groups with very small(More)
We say that a bipartite graph Γ(V 1 ∪ V 2 , E) has bi-degree r, s if every vertex from V 1 has degree r and every vertex from V 2 has degree s. Γ is called an (r, s, t)–graph if, additionally, the girth of Γ is 2t. For t > 3, very few examples of (r, s, t)–graphs were previously known. In this paper we give a recursive construction of (r, s, t)–graphs for(More)