# Uniform decompositions of complete multigraphs

We examine the notion of uniformity in the context of decompositions of complete multigraphs. A decomposition D_1,D_2,...,D_r of a graph is uniform if there exists some graph H such that D_i\cup D_j is isomorphic to H whenever 1\leq i<j\leq r. Outside of 1-factorisations, questions concerning the existence of uniform graph decompositions have not been considered previously. We prove that uniform m-cycle decompositions of the complete multigraph \mu K_n exist only when m=n, when \mu=2 and m=n-1, when \mu=1 and m=(n-1)/2, and possibly when \mu=1 and 2m(m+1)=n(n-1). If time permits, some results on uniform decompositions into graphs other than cycles may also be presented

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We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

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