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N n-1 ; 2 and n n-2 ; 2, respectively. in a paper with susan marshall [62] he showed that the conjecture is true if n is divisible by 4 and congruent modulo m to some k with m 2 k 1989, hoffman, lindner, and rodger [j. graph theory 13 1989 ; , 417426] had developed the so-called reduction step, showing that in the case with m and n both odd, it is sufficient to prove the conjecture for n in the interval [m, 3m ; . relying on this result and working with heather gavlas, brian discovered a new approach for constructing mcycle decompositions of complete graphs of order n in the case where m and n are both odd, and nearly complete graphs of order n in the case where m and n are both even, thus completely solving the problem for these two of the four cases in [72]. further extending brian's powerful techniques, one of his students, mateja sajna, completed the proof of the conjecture for the remaining two cases with m and n of opposite parity. following these break-through results, the focus of research in this area shifted to other problems, including decompositions of complete and nearly complete graphs into cycles of unequal length. to this problem brian contributed a profound conjecture [research problem 3, discrete math. 36 1981 ; , 333334] as early as 1981; namely, that kn for n odd, and kn - i for n even, can be decomposed into cycles of lengths m1 , m2 , . and only if the sum m1 + m2 the cycle lengths equals the total number of edges. this conjecture remains wide open to this day, although it has been proved in many special cases and no counterexamples have been found. recently, working with heather gavlas, mateja sajna, and helen verrall, brian was able to extend his pioneering techniques for constructing cycle decompositions of complete and nearly complete graphs to complete symmetric digraphs [74], thus proving a conjecture of bermond and faber, which had been open since 1976. this conjecture claimed that the complete symmetric digraph on n vertices with one arc in each direction between each pair of vertices ; can be decomposed into directed cycles of length m if and only if the obvious necessary condition that the cycle length m divide the number of arcs n n - 1 ; satisfied, except in the case that the pair m, n ; is one of 3, 6 ; , 4, ; , and 6, ; , in which case such a decomposition does not exist. despite much effort invested into proving this conjecture over the years, only partial results had been known before brian tackled the problem. the paper [74] received high acclaim, and was for a while featured on the jcta web page as one of the most downloaded articles.
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Before it is returned, the resulting presentation is being simplified by appropriate calls of the function simplifypresentation see 46.7 ; , but without allowing any eliminations of generators. this restriction guarantees that we get a bijection between the list of generators of g and the list of generators in the presentation. hence, if the generators of g are redundant and if you don't care for the bijection, you may get a shorter presentation by calling the function simplifypresentation, now without this restriction, once more yourself. gap h : group [ 2, 5, 3 ; , 2, 7, 5 ; , 1, 8, gap p : presentationviacosettable h presentation with 6 gens and 12 rels of total length 42 gap simplifypresentation p #i there are 4 generators and 10 relators of total length 36 if you apply the function fpgrouppresentation to the resulting presentation you will get a finitely presented group isomorphic to g. note, however, that the function isomorphismfpgroup see 45.10.1 ; is recommended for this purpose.
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