ALGEBRA AND COMBINATORICS SEMINAR Tuesday 1 April 2008, 3pm in Room 67-641 Daniel Horsley, UQ Cycle Decompositions of Complete Graphs Abstract In 1981 Alspach asked when a complete graph of order n can be decomposed into edge-disjoint cycles of specified lengths m1, m2, ... , mt. The answer to this question seems to be ``whenever there's no obvious reason that it can't be done'', but so far no-one has managed to prove this in general. Darryn Bryant and I have been working at this problem for some time and have proved two lemmas which allow us to get new cycle decompositions of complete graphs from existing cycle decompositions of complete graphs. We can use these lemmas to show that to answer Alspach's problem in general, it suffices to answer it only for lists of cycle lengths which satisfy certain (very restrictive) properties. In this talk I will give a description of these two lemmas and our reduction the problem. I will also indicate how we believe we can use our reduction of the problem to answer Alspach's problem for complete graphs of large odd order. All welcome. ----------------------------------------------------------- http://www.maths.uq.edu.au/cdmc/Seminars.html